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generalized minimal (probable) primes
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There are researches for minimal primes in base b: [URL]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL], and data for minimal primes and remaining families in bases 2 to 30: [URL]https://github.com/curtisbright/mepn-data/tree/master/data[/URL], data for minimal primes and remaining families in bases 28 to 50: [URL]https://github.com/RaymondDevillers/primes[/URL].
This is a text file for minimal primes in bases 2 to 16. |
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These are unsolved families in base 2 to base 36, given by the links. (see the links for the top (probable) primes)
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Is anyone reserving these families?
Bases <= 36 with only few families remaining: Base 17: F1{9}: (4105*17^n-9)/16 Base 19: EE1{6}: (15964*19^n-1)/3 Base 21: G{0}FK: 7056*21^n+335 Base 26: {A}6F: (1352*26^n-497)/5 {I}GL: (12168*26^n-1243)/25 Base 28: O{A}F: (18424*28^n+125)/27 Base 36: O{L}Z: (4428*36^n+67)/5 {P}SZ: (6480*36^n+821)/7 |
The letters A, B, C, D, ... are the digits: A=10, B=11, C=12, D=13, ...
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The status:
Base 17: F1{9}: (4105*17^n-9)/16: at n=1M, no (probable) prime found. Base 19: EE1{6}: (15964*19^n-1)/3: at n=707K, no (probable) prime found. Base 21: G{0}FK: 7056*21^n+335: at n=506K, no (probable) prime found. Base 25: EF{O}: 366*25^n-1: at n=660K, no (probable) prime found. OL{8}: (4975*25^n-111)/8: at n=303K, no (probable) prime found. CM{1}: (7729*25^n-1)/24: at n=303K, no (probable) prime found. E{1}E: (8425*25^n+311)/24: at n=303K, no (probable) prime found. ... (all other unsolved families in base 25 may be tested to n=303K) Base 26: {A}6F: (1352*26^n-497)/5: at n=486K, no (probable) prime found. {I}GL: (12168*26^n-1243)/25: at n=497K, no (probable) prime found. Base 27: 8{0}9A: 5832*27^n+253: at n=368K, no (probable) prime found. C{L}E: (8991*27^n-203)/26: at n=368K, no (probable) prime found. 999{G}: (88577*27^n-8)/13: at n=368K, no (probable) prime found. E{I}F8: (139239*27^n-1192)/13: at n=368K, no (probable) prime found. {F}9FM: (295245*27^n-113557)/26: at n=368K, no (probable) prime found. Base 28: O{A}F: (18424*28^n+125)/27: at n=543K, no (probable) prime found. Base 29: All the unsolved families may be tested to n=242K. Since the page [URL]https://github.com/curtisbright/mepn-data/tree/master/data[/URL] only solve the minimal prime problem to bases b<=30, for 31<=b<=36, these bases are reserved by me (these bases have already reserve to n=10K). Now, I am reserving bases 31, 35 and 36, use factordb. In fact, I decide to solve the minimal prime problem to all bases b<=64 in the future. However, at present, I only solve this problem to all bases b<=36. I will reserve bases 37<=b<=64 if all the bases b<=36 have been tested to at least n=1M. |
For the two unsolved families in base 36:
O{L}Z: (30996*36^n+469)/35: tested up to n=15815, no (probable) prime found. {P}SZ: (6480*36^n+821)/7: tested up to n=15815, no (probable) prime found. |
Base 31:
E8{U}P: 13733*31^n-6: at n=15K, no (probable) prime found. {P}I: (155*31^n-47)/6: at n=15K, no (probable) prime found. {R}1: (279*31^n-269)/10: at n=15K, no (probable) prime found. {U}P8K: 29791*31^n-5498: at n=15K, no (probable) prime found. |
These problems are to find a prime of the form (k*b^n+c)/gcd(k+c,b-1) with integer n>=1 for fixed integers k, b and c, k>=1, b>=2, gcd(k,c)=1 and gcd(b,c)=1.
For some (k,b,c), there cannot be any prime because of covering set (e.g. (k,b,c) = (78557,2,1), (334,10,-1) or (84687,6,-1)) or full algebra factors (e.g. (k,b,c) = (9,4,-1), (2500,16,1) or (9,4,-25) (the case (9,4,-25) can produce prime [I]only[/I] for n=1)) or partial algebra factors (e.g. (k,b,c) = (144,28,-1), (25,17,-9) or (1369,30,-1)). It is conjectured that for every (k,b,c) which cannot be proven that they do not have any prime, there are infinitely primes of the form (k*b^n+c)/gcd(k+c,b-1). (Notice the special case: (k,b,c) = (8,128,1), it cannot have any prime but have neither covering set nor algebra factors) However, there are many such cases even not have a single known prime, like (21181,2,1), (2293,2,-1), (4,53,1), (1,185,-1), (1,38,1), (269,10,1), (197,7,-1), (4105,17,-9), (16,21,335), (5,36,821), but not all case will produce a minimal prime to base b, e.g. the form (197*7^n-1)/2 is the form 200{3} in base 7, but since 2 is already prime, the smallest prime of this form (if exists) will not be a minimal prime in base 7. The c=1 and gcd(k+c,b-1)=1 case is the Sierpinski problem base b, and the c=-1 and gcd(k+c,b-1)=1 case is the Riesel problem base b. |
Other special cases:
k=1, c=1, b is even: the generalized Fermat primes in base b. k=1, c=1, b is odd: the generalized half Fermat primes in base b. k=1, c=-1: the repunit primes in base b. |
Also,
k=1, c>0: the dual Sierpinski problem base b. k=1, c<0: the dual Riesel problem base b. |
(k*b^n+c)/gcd(k+c,b-1) has full algebra factors if and only if at least one of the following conditions holds:
* There is an integer r>1 such that k, b and -c are all perfect r-th powers. or * b and 4kc are both perfect 4th powers. |
[QUOTE=sweety439;458107]These problems are to find a prime of the form (k*b^n+c)/gcd(k+c,b-1) with integer n>=1 for fixed integers k, b and c, k>=1, b>=2, gcd(k,c)=1 and gcd(b,c)=1.
For some (k,b,c), there cannot be any prime because of covering set (e.g. (k,b,c) = (78557,2,1), (334,10,-1) or (84687,6,-1)) or full algebra factors (e.g. (k,b,c) = (9,4,-1), (2500,16,1) or (9,4,-25) (the case (9,4,-25) can produce prime [I]only[/I] for n=1)) or partial algebra factors (e.g. (k,b,c) = (144,28,-1), (25,17,-9) or (1369,30,-1)). It is conjectured that for every (k,b,c) which cannot be proven that they do not have any prime, there are infinitely primes of the form (k*b^n+c)/gcd(k+c,b-1). (Notice the special case: (k,b,c) = (8,128,1), it cannot have any prime but have neither covering set nor algebra factors) However, there are many such cases even not have a single known prime, like (21181,2,1), (2293,2,-1), (4,53,1), (1,185,-1), (1,38,1), (269,10,1), (197,7,-1), (4105,17,-9), (16,21,335), (5,36,821), but not all case will produce a minimal prime to base b, e.g. the form (197*7^n-1)/2 is the form 200{3} in base 7, but since 2 is already prime, the smallest prime of this form (if exists) will not be a minimal prime in base 7. The c=1 and gcd(k+c,b-1)=1 case is the Sierpinski problem base b, and the c=-1 and gcd(k+c,b-1)=1 case is the Riesel problem base b.[/QUOTE] gcd(k+c,b-1) is the largest number that divides k*b^n+c for all n. Note: gcd(0, m) = m for all integer m, and gcd(1, m) = 1 for all integer m. |
[QUOTE=sweety439;459400]Note: gcd(0, m) = m for all integer m, and gcd(1, m) = 1 for all integer m.[/QUOTE]
Thank you, Captain Obvious! In other news today, [SPOILER]light travels faster than sound, and a minute contains 60 seconds.[/SPOILER] |
[QUOTE=Batalov;459411][SPOILER]…and a minute contains 60 seconds.[/SPOILER][/QUOTE]Is that always true?
[SPOILER]https://en.wikipedia.org/wiki/Leap_second[/SPOILER] :mike: |
[QUOTE=Xyzzy;459441]Is that always true?
[/QUOTE] It is just as true as "[I]gcd(0, m) = m for all integer m[/I]". [SPOILER]For m=0, gcd(0, 0) = 24. 24 is a greatest common divisor of 0 and 0, because it divides both 0 and 0, and there is no higher number: see goo.gl/ASN4Ov ... and 24 ≠ 0[/SPOILER] |
[QUOTE=Batalov;459443]It is just as true as "[I]gcd(0, m) = m for all integer m[/I]".
[SPOILER]For m=0, gcd(0, 0) = 24. 24 is a greatest common divisor of 0 and 0, because it divides both 0 and 0, and there is no higher number: see goo.gl/ASN4Ov ... and 24 ≠ 0[/SPOILER][/QUOTE] No, gcd(0, m) = m is true only for [I][B]positive[/B][/I] integer m. :smile: |
[QUOTE=sweety439;459463]No, gcd(0, m) = m is true only for [I][B]positive[/B][/I] integer m. :smile:[/QUOTE]
gcd(0, 0) = 0 so the rule holds in all cases [SPOILER]except when you want to take 24 as maximal instead of 0 for humor[/SPOILER]. |
[QUOTE=Batalov;459411][spoiler]light travels faster than sound[/spoiler][/QUOTE]
Depends on the medium. Through a vacuum, it certainly does. Also through the air we breathe. But it takes a long, long time for the EM energy produced in the solar core to make its way through the interior of the sun, and out as sunshine. Sound waves travel through the interior of the sun much more quickly. |
[QUOTE=Dr Sardonicus;459519]Depends on the medium. Through a vacuum, it certainly does. Also through the air we breathe. But it takes a long, long time for the EM energy produced in the solar core to make its way through the interior of the sun, and out as sunshine. Sound waves travel through the interior of the sun much more quickly.[/QUOTE]
If the medium is not transparent, the the speed of (visible light) is zero, thus it is lower then that of sound. Besides, if the medium is vacuum, then the speed of sound is zero, since sound needs medium to spread. |
[QUOTE=Batalov;459411]Thank you, Captain Obvious!
In other news today, [SPOILER]light travels faster than sound, and a minute contains 60 seconds.[/SPOILER][/QUOTE] A minute does not always contain 60 seconds, since the definition of second is from the cesium atomic, it is not always 1/60 minute = 1/86400 day, since the definition of day is form earth. Besides, [SPOILER]in Alaska and in Amazon forest, the length of "second" is not the same, since the distance of them to geocentric is different XDDD...[/SPOILER] |
Base 36 has only two unsolved family:
(4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3 |
[QUOTE=sweety439;531551]Base 36 has only two unsolved family:
(4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3[/QUOTE] A (probable) prime was found: (13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime" |
[QUOTE=sweety439;531552]A (probable) prime was found:
(13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime"[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) currently at n=21939, no (probable) prime found. (4428*36^n+67)/5 (O{L}Z in base 36) currently at n=23729, no (probable) prime found. (6480*36^n+821)/7 ({P}SZ in base 36) currently at n=20235, no (probable) prime found. |
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[QUOTE=sweety439;531553](86*40^n+37)/3 (S{Q}d in base 40) currently at n=21939, no (probable) prime found.
(4428*36^n+67)/5 (O{L}Z in base 36) currently at n=23729, no (probable) prime found. (6480*36^n+821)/7 ({P}SZ in base 36) currently at n=20235, no (probable) prime found.[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) tested to n=25K, no (probable) prime found. Extended to n=50K |
(4428*36^n+67)/5 (O{L}Z in base 36) currently at n=32401, no (probable) prime found.
(6480*36^n+821)/7 ({P}SZ in base 36) currently at n=26743, no (probable) prime found. |
[QUOTE=sweety439;531552]A (probable) prime was found:
(13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime"[/QUOTE] See the page [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] |
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[QUOTE=sweety439;531599](86*40^n+37)/3 (S{Q}d in base 40) tested to n=25K, no (probable) prime found.
Extended to n=50K[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) seems to have a low weight, for 25K<=n<=50K, sieve to p=10^9, only 481 n remain. |
[QUOTE=sweety439;531600](4428*36^n+67)/5 (O{L}Z in base 36) currently at n=32401, no (probable) prime found.
(6480*36^n+821)/7 ({P}SZ in base 36) currently at n=26743, no (probable) prime found.[/QUOTE] I know that they can be reduced to (123*36^n+67)/5 and (5*36^n+821)/7, however, we let n be the number of the digits in "{}" (thus, the base 40 unsolved family should be (3440*40^n+37)/3 .... |
We assume the conjecture in post [URL="https://mersenneforum.org/showpost.php?p=529838&postcount=675"]https://mersenneforum.org/showpost.php?p=529838&postcount=675[/URL] is true (thus, all families in the files "unsolved xx" in [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]https://github.com/curtisbright/mepn-data/tree/master/data[/URL] and all families in the files "left xx" in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] have infinitely many primes)
Then the number of base n digits of the largest base n minimal prime is about 2^eulerphi(n) [CODE] n length of the largest minimal prime in base n 2 2 3 3 4 2 5 5 6 5 7 5 8 9 9 4 10 8 11 45 12 8 13 32021 (PRP) 14 86 15 107 16 3545 18 33 20 449 22 764 23 800874 (PRP) 24 100 30 1024 42 487 [/CODE] [CODE] n excepted length of the largest minimal prime in base n 2 2 3 4 4 4 5 16 6 4 7 64 8 16 9 64 10 16 11 1024 12 16 13 4096 14 64 15 256 16 256 17 65536 18 64 19 262144 20 256 21 4096 22 1024 23 4194304 24 256 25 1048576 26 4096 27 262144 28 4096 29 268435456 30 256 31 1073741824 32 65536 33 1048576 34 65536 35 16777216 36 4096 37 68719476736 38 262144 39 16777216 40 65536 41 1099511627776 42 4096 43 4398046511104 44 1048576 45 16777216 46 4194304 47 70368744177664 48 65536 49 4398046511104 50 1048576 51 4294967296 52 16777216 53 4503599627370496 54 262144 55 1099511627776 56 16777216 57 68719476736 58 268435456 59 288230376151711744 60 65536 61 1152921504606846976 62 1073741824 63 68719476736 64 4294967296 65 281474976710656 66 1048576 67 73786976294838206464 68 4294967296 69 17592186044416 70 16777216 71 1180591620717411303424 72 16777216 [/CODE] |
Also, assume the conjecture in post [URL="https://mersenneforum.org/showpost.php?p=529838&postcount=675"]https://mersenneforum.org/showpost.php?p=529838&postcount=675[/URL] is true:
[CODE] n length of largest minimal prime in base n 17 >1000000 19 >707000 21 >506700 25 >660000 (because of the EF{O} family, given by [URL="https://github.com/curtisbright/mepn-data/blob/master/data/sieve.25.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/sieve.25.txt[/URL]) 26 >486700 27 >368000 28 >543000 29 >242300 31 >=524288 (because of the {F}G family, given by [URL="https://oeis.org/A275530"]https://oeis.org/A275530[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]) 32 >=3435973837 (because of the G{0}1 family, given by [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]) 33 >10000 34 >10000 35 >10000 36 >32401 (the only two unsolved families are both reserved by me) 37 >=22023 (because of the prime FY{a[SUB]22021[/SUB]}, given by CRUS) 38 >=16777217 (because of the 1{0}1 family, see [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL] and [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]) 39 >10000 40 >25000 (the only one unsolved family is reserved by me) 41 >10000 43 >10000 44 >10000 45 >=18523 (because of the prime O{0[SUB]18521[/SUB]}1, given by CRUS, note that the prime AO{0[SUB]44790[/SUB]}1 is not a minimal prime in base 45, although AO{0}1 is in [URL="https://github.com/RaymondDevillers/primes/blob/master/left45"]https://github.com/RaymondDevillers/primes/blob/master/left45[/URL]) 46 >250000 (because of the d4{0}1 family, given by CRUS) 47 >10000 48 >250000 (because of the a{0}1 family, given by CRUS) 49 >=52700 (because of the prime SL{m[SUB]52698[/SUB]}, given by CRUS) 50 >=16777217 (because of the 1{0}1 family, see [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL] and [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]) [/CODE] |
[QUOTE=sweety439;531551]Base 36 has only two unsolved family:
(4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3[/QUOTE] The two unsolved family should be: (559920*40^n+29)/13 (3440*40^n+37)/3 and this (probable) prime should be: (559920*40^12380+29)/13 (13998*40^12381+29)/13 is the reduced form |
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No (probable) prime found for (86*40^n+37)/3 (S{Q}d in base 40) for n=25K-50K.
Text file attached. Extended to n=100K. |
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Base 36:
O{L}Z (4428*36^n+67)/5: tested to n=50K, no (probable) prime found {P}SZ (6480*36^n+821)/7: currently at n=41566, no (probable) prime found Base 40: S{Q}d (86*40^n+37)/3: currently at n=59777, no (probable) prime found |
[QUOTE=sweety439;531552]A (probable) prime was found:
(13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime"[/QUOTE] Another probable prime is (13998*40^13474+29)/13, but this is not minimal prime in base 40 |
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[QUOTE=sweety439;531730]Base 36:
O{L}Z (4428*36^n+67)/5: tested to n=50K, no (probable) prime found {P}SZ (6480*36^n+821)/7: currently at n=41566, no (probable) prime found Base 40: S{Q}d (86*40^n+37)/3: currently at n=59777, no (probable) prime found[/QUOTE] Base 36 {P}SZ (6480*36^n+821)/7 tested to n=50K, no (probable) prime found. Result file attached. |
Base 40:
S{Q}d (86*40^n+37)/3: currently at n=87437, no (probable) prime found |
Unsolved families:
Base 17: F1{9}: (4105*17^n-9)/16 Base 19: EE1{6}: (15964*19^n-1)/3 Base 21: G{0}FK: 7056*21^n+335 Base 25: EF{O}: 366*25^n-1 O{L}8: (4975*25^n-111)/8 CM{1}: (7729*25^n-1)/24 E{1}E: (8425*25^n+311)/24 EE{1}: (8737*25^n-1)/24 6M{F}9: (34525*25^n-53)/8 F{1}F1: (225625*25^n+8399)/24 Base 26: {A}6F: (1352*26^n-497)/5 {I}GL: (12168*26^n-1243)/25 Base 31: E8{U}P: 13733*31^n-6 {F}RA: (961*31^n+733)/2 {F}G: (31*31^n+1)/2 {F}KO: (961*31^n+327)/2 IE{L}: (5727*31^n-7)/10 {L}G: (217*31^n-57)/10 {L}CE: (6727*31^n-2867)/10 M{P}: (137*31^n-5)/6 {P}I: (155*31^n-47)/6 {R}1: (279*31^n-269)/10 {R}8: (279*31^n-199)/10 {U}P8K: 29791*31^n-5498 |
Base 31:
ILE{L}: (179637*31^n-7)/10 [need not to be searched if a smaller prime for the "IE{L}: (5727*31^n-7)/10" family were found] L{F}G: (1333*31^n+1)/2 [need not to be searched if a smaller prime for the "{F}G: (31*31^n+1)/2" family were found] L0{F}G: (40393*31^n+1)/2 [need not to be searched if a smaller prime for either the "{F}G: (31*31^n+1)/2" family or the "L{F}G: (1333*31^n+1)/2" family were found] {L}9G: (6727*31^n-3777)/10 [need not to be searched if a smaller prime for the "{L}G: (217*31^n-57)/10" family were found] {L}9IG: (208537*31^n-116307)/10 [need not to be searched if a smaller prime for either the "{L}G: (217*31^n-57)/10" family or the "{L}9G: (6727*31^n-3777)/10" family were found] {L}SO: (6727*31^n+2193)/10 {L}IS: (6727*31^n-867)/10 MI{O}L: (108624*31^n-19)/5 P{F}G: (1581*31^n+1)/2 [need not to be searched if a smaller prime for the "{F}G: (31*31^n+1)/2" family were found] PEO{0}Q: 758973*31^n+26 {R}1R: (8649*31^n-8069)/10 [need not to be searched if a smaller prime for the "{R}1: (279*31^n-269)/10" family were found] SP{0}K: 27683*31^n+20 |
Base 35:
6W{P}4: (288855*35^n-739)/34 [need not to be searched if a smaller prime for the "W{P}4: (38955*35^n-739)/34" family were found] F8{0}F9: 652925*35^n+534 {Y}PO: 1225*35^n-326 FQ{F}I: (656215*35^n+87)/34 PX{0}ER: 1112300*35^n+517 Q{P}4: (31815*35^n-739)/34 RF{0}CPI: 41160000*35^n+15593 Base 36: O{L}Z: (30996*36^n+469)/35 {P}SZ: (6480*36^n+821)/7 Base 40: S{Q}d: (3440*40^n+37)/3 |
Base 25:
F{O}KO: 9375*25^n+524 FO{K}O: (56375*25^n+19)/6 LO{L}8: (109975*25^n-111)/8 [need not to be searched if a smaller prime for the "O{L}8: (4975*25^n-111)/8" family were found] M{1}F1: (330625*25^n+8399)/24 M1{0}8: 13775*25^n+8 Base 28: O{A}F: (18424*28^n+125)/27 Base 35: LAA{E}6: (15520820*35^n-143)/17 {L}E6: (25725*35^n-8861)/34 P0{P}G: (1042125*35^n-331)/34 {Q}PEM: (557375*35^n-28046)/17 RU{A}C: (580300*35^n+29)/17 W{P}4: (38955*35^n-739)/34 {X}MLX: (1414875*35^n-472463)/34 X{M}Y: (20020*35^n+193)/17 |
Base 27:
8{0}9A: 5832*27^n+253 999{G}: (88577*27^n-8)/13 C{L}E: (8991*27^n-203)/26 E{I}F8: (139239*27^n-1192)/13 {F}9FM: (295245*27^n-113557)/26 Base 48: A{0}SP: 23040*48^n+1369 C{e}Z: (28992*48^n-275)/47 {K}IP: (46080*48^n-4297)/47 a{0}1: 1728*48^n+1 eL{0}Z: 93168*48^n+35 jc{e}Z: (4960608*48^n-275)/47 |
The "minimal prime problem" is solved only in bases 2~16, 18, 20, 22~24, 30, 42, and maybe 60
[CODE] b, length of largest minimal prime base b, number of minimal primes base b 2, 2, 2 3, 3, 3 4, 2, 3 5, 5, 8 6, 5, 7 7, 5, 9 8, 9, 15 9, 4, 12 10, 8, 26 11, 45, 152 12, 8, 17 13, 32021, 228 14, 86, 240 15, 107, 100 16, 3545, 483 18, 33, 50 20, 449, 651 22, 764, 1242 23, 800874, 6021 24, 100, 306 30, 1024, 220 42, 487, 4551 60, 1938, ? [/CODE] |
Some minimal (probable) primes with bases 28<=b<=50 not shown in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL]: (and hence some unsolved families can be removed)
Base 37: (families FY{a} and R8{a} can be removed) 590*37^22021-1 (= FY{a_22021}) 1008*37^20895-1 (= R8{a_20895}) Base 40: (family Qa{U}X can be removed) (13998*40^12381+29)/13 (= Qa{U_12380}X) Base 45: (families O{0}1 and AO{0}1 can be removed, and hence families O{0}1F1, O{0}ZZ1, unless they have small (probable) primes) 24*45^18522+1 (= O{0_18521}1) 474*45^44791+1 (= AO{0_44790}1) [this prime is not minimal prime] Base 49: (families 11c{0}1, Fd{0}1, SL{m} and Yd{m} can be removed, and hence families S6L{m}, YUUd{m}, YUd{m}, unless they have small (probable) primes) 2488*49^29737+1 (= 11c{0_29736}1) 774*49^18341+1 (= Fd{0_18340}1) 1394*49^52698-1 (= SL{m_52698}) 1706*49^16337-1 (= Yd{m_16337}) |
Although the test limit of all families in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] are all 10K, but some families are in fact already tested to much higher....
Base 25: EF{O}, 366*25^n-1: 260K, see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL] Base 31: F{G}, (1*31^n+1)/2: 2^19-2, see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL] Base 32: 4{0}1, 4*32^n+1: (2^33-7)/5, see [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL] G{0}1, 16*32^n+1: (2^34-9)/5, see [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL] UG{0}1, 976*32^n+1: 560K, see [URL="http://www.prothsearch.com/riesel1.html"]http://www.prothsearch.com/riesel1.html[/URL] Base 38: 1{0}1 1*38^n+1: 2^24-2, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL] Base 45: 9W1{0}1 19666*45^n+1: 100K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] Base 46: d4{0}1, 1798*46^n+1: 500K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] Base 48: a{0}1, 36*48^n+1: 500K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] Base 50: 1{0}1 1*50^n+1: 2^24-2, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL] |
Unsolved families which are CRUS Sierpinski/Riesel problems but with k's > CK:
Base 32: G{0}1: 16*32^n+1 UG{0}1: 976*32^n+1 Base 33: FFF{W}: 16846*33^n-1 Base 41: FZ{0}1: 650*41^n+1 R0R8{0}1: 1861982*41^n+1 S{0}1: 28*41^n+1 XL4{0}1: 56338*41^n+1 Z098{0}1: 2412612*41^n+1 Z0R{0}1: 58862*41^n+1 EF{e}: 590*41^n-1 PI{e}: 1044*41^n-1 UFM{e}: 51068*41^n-1 UX{e}: 1264*41^n-1 XOC{e}: 56470*41^n-1 XQO{e}: 56564*41^n-1 XQXXXX{e}: 3899055672*41^n-1 XQ{e}: 1380*41^n-1 Base 43: Y6{0}1: 1468*43^n+1 6XF{0}1: 12528*43^n+1 8Q6{0}1: 15916*43^n+1 XZZ{g}: 62558*43^n-1 YFa{g}: 63548*43^n-1 dcU{g}: 73776*43^n-1 4ZZZ{g}: 384284*43^n-1 8OR{g}: 15852*43^n-1 9QQ{g}: 17786*43^n-1 FFFFFFFQ{g}: 4174357242012*43^n-1 FFFQ{g}: 1221012*43^n-1 |
All smallest generalized repunit prime base b are minimal prime base b, since they are of the form {1} in base b, for the smallest generalized repunit (probable) prime base b for b<=1024, see [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]
All smallest generalized Fermat prime base b (for even b) and all smallest generalized half Fermat prime base b (for odd b) are minimal prime base b, unless (b-1)/2 is prime for odd b, since they are of the form 1{0}1 in base b (for even b) or {(b-1)/2}(b+1)/2 in base b (for odd b), for the smallest generalized (half) Fermat (probable) prime base b for b<=1024, see [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Sierp%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Sierp%20k1.txt[/URL] |
There are no known generalized repunit (probable) primes in these bases <= 1024: (search limit: 100000)
{185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015} There are no known generalized (half) Fermat (probable) primes in these bases <= 1024: (search limit: 2^22 for GFN for even bases, 2^18 for half GFN for odd bases) {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016} |
[QUOTE=sweety439;561623]There are no known generalized repunit (probable) primes in these bases <= 1024: (search limit: 100000)
{185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015} There are no known generalized (half) Fermat (probable) primes in these bases <= 1024: (search limit: 2^22 for GFN for even bases, 2^18 for half GFN for odd bases) {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016}[/QUOTE] The GFN for these bases (always minimal primes): {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016} The half GFN for these bases are also minimal primes: {31, 55, 67, 77, 89, 91, 97, 99, 109, 127, 137, 149, 151, 155, 161, 183, 189, 197, 211, 223, 233, 235, 241, 247, 249, 257, 265, 269, 281, 283, 285, 287, 291, 293, 307, 311, 319, 351, 355, 367, 369, 377, 389, 401, 407, 411, 413, 417, 437, 439, 443, 469, 473, 475, 489, 493, 495, 497, 509, 511, 533, 547, 549, 591, 593, 597, 601, 603, 611, 619, 629, 637, 645, 647, 651, 653, 655, 659, 667, 671, 679, 683, 687, 691, 709, 731, 737, 741, 743, 749, 753, 755, 771, 773, 775, 783, 785, 787, 797, 807, 809, 813, 823, 825, 849, 851, 853, 873, 889, 893, 903, 907, 911, 937, 939, 941, 943, 945, 947, 953, 961, 967, 977, 987, 993, 1003, 1009} However, the half GFN for these bases are not minimal primes, since (b-1)/2 is prime: {63, 83, 107, 123, 135, 143, 147, 179, 207, 215, 227, 255, 263, 303, 327, 347, 359, 383, 387, 423, 447, 467, 483, 515, 555, 563, 615, 627, 635, 663, 675, 707, 735, 759, 767, 867, 887, 899, 915, 923, 927, 975, 983, 999} |
The remain k < b (also including k > CK, if k < b, e.g. 28*41^n+1 and 27*34^n-1) in CRUS corresponding to minimal primes to base b if and only if:
* In Sierpinski case, k is not prime (since it is k{0}1 in base b) * In Riesel case, neither k-1 nor b-1 is prime (since it is (k-1){(b-1)} in base b) e.g. the smallest prime of the form 4*53^n+1 (already searched to 1.65M) will be minimal prime base 53, if it exists (CRUS conjectured that they all exist) |
[QUOTE=sweety439;561624]The GFN for these bases (always minimal primes):
{38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016} The half GFN for these bases are also minimal primes: {31, 55, 67, 77, 89, 91, 97, 99, 109, 127, 137, 149, 151, 155, 161, 183, 189, 197, 211, 223, 233, 235, 241, 247, 249, 257, 265, 269, 281, 283, 285, 287, 291, 293, 307, 311, 319, 351, 355, 367, 369, 377, 389, 401, 407, 411, 413, 417, 437, 439, 443, 469, 473, 475, 489, 493, 495, 497, 509, 511, 533, 547, 549, 591, 593, 597, 601, 603, 611, 619, 629, 637, 645, 647, 651, 653, 655, 659, 667, 671, 679, 683, 687, 691, 709, 731, 737, 741, 743, 749, 753, 755, 771, 773, 775, 783, 785, 787, 797, 807, 809, 813, 823, 825, 849, 851, 853, 873, 889, 893, 903, 907, 911, 937, 939, 941, 943, 945, 947, 953, 961, 967, 977, 987, 993, 1003, 1009} However, the half GFN for these bases are not minimal primes, since (b-1)/2 is prime: {63, 83, 107, 123, 135, 143, 147, 179, 207, 215, 227, 255, 263, 303, 327, 347, 359, 383, 387, 423, 447, 467, 483, 515, 555, 563, 615, 627, 635, 663, 675, 707, 735, 759, 767, 867, 887, 899, 915, 923, 927, 975, 983, 999}[/QUOTE] These bases are the bases <= 1024 which is not perfect odd power (of the form m^r with odd r>1) whose "minimal prime program" have GFN or half GFN remain, for the bases <= 1024 which is perfect odd power (of the form m^r with odd r>1): * Cubes: ** Base 8: GFN in base 2 are either 2{0}1 or 4{0}1 in base 8, however, 2 and 401 are primes, thus, base 8 does not have GFN or half GFN remain. ** Base 27: half GFN in base 3 are either 1{D}E or 4{D}E in base 27, however, D is prime, thus, base 27 does not have GFN or half GFN remain. ** Base 64: GFN in base 2 are either 4{0}1 or G{0}1 in base 64, however, 41 and G01 are primes, thus, base 64 does not have GFN or half GFN remain. ** Base 125: half GFN in base 5 are either 2:{62}:63 or 12:{62}:63 in base 125, however, 2 is prime, but the family 12:{62}:63 does not have any known (probable) prime (the only known half GFN (probable) primes in base 5 are 3, 13, 2:63), thus, [B][I]base 125 has half GFN remain.[/I][/B] ** Base 216: GFN in base 6 are either 6:{0}:1 or 36:{0}:1 in base 216, however, 6:1 is prime, but the family 36:{0}:1 does not have any known prime (the only known GFN primes in base 6 are 7, 37, 6:1), thus, [B][I]base 216 has GFN remain.[/I][/B] ** Base 343: half GFN in base 7 are either 3:{171}:172 or 24:{171}:172 in base 343, however, 3 is prime, but the family 24:{171}:172 does not have any known (probable) prime (the only known half GFN (probable) prime in base 7 is 3:172), thus, [B][I]base 343 has half GFN remain.[/I][/B] ** Base 512: GFN in base 2 are 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 128:{0}:1, or 256:{0}:1 in base 512, however, 2 and 128:1 are primes, but the families 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 128:1), thus, [B][I]base 512 has GFN remain.[/I][/B] ** Base 729: half GFN in base 3 are either 4:{364}:365 or 40:{364}:365 in base 729, however, 40:364:365 and 4:364:364:364:364:365 are primes, thus, base 729 does not have GFN or half GFN remain. ** Base 1000: GFN in base 10 are either 10:{0}:1 or 100:{0}:1 in base 1000, and both families do not have any known prime (the only known GFN primes in base 10 are 11 and 101), thus, [B][I]base 1000 has GFN remain.[/I][/B] * 5th powers: ** Base 32: GFN in base 2 are 2{0}1, 4{0}1, 8{0}1, or G{0}1 in base 32, however, 2 and 81 are primes, but the families 4{0}1 and G{0}1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, H, 81, 2001), thus, [B][I]base 32 has GFN remain.[/I][/B] ** Base 243: half GFN in base 3 are 1:{121}:122, 4:{121}:122, 13:{121}:122, or 40:{121}:122 in base 243, however, 1:121:121:122, 4:121, 13, 40:121:121:121:121:121:121:121:121:121:121:121:122 are primes, thus, base 243 does not have GFN or half GFN remain. ** Base 1024: GFN in base 2 are 4:{0}:1, 16:{0}:1, 64:{0}:1, or 256:{0}:1 in base 1024, however, 64:1 is prime, but the families 4:{0}:1, 16:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 64:1), thus, [B][I]base 1024 has GFN remain.[/I][/B] * 7th powers: ** Base 128: GFN in base 2 are 2:{0}:1, 4:{0}:1, or 16:{0}:1 in base 128, however, 2 and 4:0:1 are primes, but the family 16:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 2:1, 4:0:1), thus, [B][I]base 128 has GFN remain.[/I][/B] |
There are about exp(gamma*k) minimal primes in base n, where k = number of 2-digit numbers [I]xy[/I] in base n such that none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1
where exp(x) = e^x (e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] (2.718281828...), gamma is [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant"]Euler–Mascheroni constant[/URL] (0.5772156649...)) Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), where k = number of 2-digit numbers [I]xy[/I] in base n such that [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1 |
[QUOTE=sweety439;562748]There are about exp(gamma*k) minimal primes in base n, where k = number of 2-digit numbers [I]xy[/I] in base n such that none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1
where exp(x) = e^x (e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] (2.718281828...), gamma is [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant"]Euler–Mascheroni constant[/URL] (0.5772156649...)) Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), where k = number of 2-digit numbers [I]xy[/I] in base n such that [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1[/QUOTE] The reason is if and only if a 2-digit number [I]xy[/I] satisfies all these condition (none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1), then [I]xy[/I] can be the first and last digit of a "base n minimal prime" with >=3 digits (if we require the primes have >=2 digits, then the conditions [I]x[/I] is not prime, [I]y[/I] is not prime, are both not needed, only need [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1) The k for the original case (i.e. including the single-digit primes) [CODE] base,k 2,0 3,1 4,0 5,4 6,1 7,5 8,3 9,8 10,5 11,27 12,2 13,38 14,10 15,23 16,17 17,84 18,4 19,108 20,17 21,59 22,30 23,164 24,9 25,151 26,57 27,136 28,55 29,307 30,8 31,350 32,87 33,190 34,111 35,282 36,42 37,539 38,144 39,289 40,107 41,678 42,31 43,736 44,169 45,295 46,227 47,892 48,59 49,804 50,160 51,543 52,286 53,1194 54,85 55,842 56,284 57,731 58,416 59,1545 60,47 61,1627 62,464 63,738 64,508 65,1248 66,144 67,2031 68,537 69,1101 70,265 71,2296 72,190 73,2404 74,676 75,936 76,696 77,1943 78,203 79,2867 80,503 81,1623 82,912 83,3179 84,150 85,2275 86,999 87,1865 88,911 89,3750 90,110 91,2865 92,1121 93,2182 94,1285 95,3009 96,456 97,4603 98,1012 99,2249 100,901 101,4994 102,420 103,5158 104,1347 105,1500 106,1635 107,5562 108,539 109,5725 110,812 111,3123 112,1300 113,6178 114,502 115,4391 116,1852 117,3231 118,2048 119,5209 120,273 121,6478 122,2286 123,4081 124,2313 125,5810 126,536 127,8241 128,2568 [/CODE] The k for the case for prime with >=2 digits: [CODE] base,k 2,0 3,2 4,2 5,10 6,2 7,25 8,14 9,30 10,15 11,75 12,15 13,111 14,40 15,70 16,72 17,202 18,43 19,260 20,82 21,163 22,126 23,394 24,88 25,375 26,187 27,348 28,196 29,648 30,88 31,749 32,335 33,470 34,348 35,627 36,221 37,1089 38,450 39,684 40,385 41,1350 42,231 43,1495 44,579 45,764 46,685 47,1802 48,425 49,1674 50,628 51,1237 52,846 53,2311 54,549 55,1742 56,891 57,1575 58,1138 59,2894 60,458 61,3099 62,1316 63,1701 64,1470 65,2512 66,724 67,3766 68,1539 69,2370 70,1021 71,4245 72,1034 73,4500 74,1927 75,2242 76,1964 77,3802 78,1076 79,5295 80,1716 81,3495 82,2395 83,5861 84,1109 85,4476 86,2654 87,3879 88,2521 89,6768 90,1142 91,5466 92,2970 93,4467 94,3202 95,5671 96,1922 97,8078 98,2914 99,4697 100,2756 101,8774 102,1984 103,9137 104,3656 105,3683 106,4130 107,9883 108,2480 109,10270 110,2942 111,6478 112,3859 113,11051 114,2551 115,8490 116,4876 117,6765 118,5170 119,9691 120,2152 121,11515 122,5547 123,8024 124,5614 125,10609 126,2682 127,14030 128,6259 [/CODE] |
[QUOTE=sweety439;562832]The reason is if and only if a 2-digit number [I]xy[/I] satisfies all these condition (none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1), then [I]xy[/I] can be the first and last digit of a "base n minimal prime" with >=3 digits (if we require the primes have >=2 digits, then the conditions [I]x[/I] is not prime, [I]y[/I] is not prime, are both not needed, only need [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1)
The k for the original case (i.e. including the single-digit primes) [CODE] base,k 2,0 3,1 4,0 5,4 6,1 7,5 8,3 9,8 10,5 11,27 12,2 13,38 14,10 15,23 16,17 17,84 18,4 19,108 20,17 21,59 22,30 23,164 24,9 25,151 26,57 27,136 28,55 29,307 30,8 31,350 32,87 33,190 34,111 35,282 36,42 37,539 38,144 39,289 40,107 41,678 42,31 43,736 44,169 45,295 46,227 47,892 48,59 49,804 50,160 51,543 52,286 53,1194 54,85 55,842 56,284 57,731 58,416 59,1545 60,47 61,1627 62,464 63,738 64,508 65,1248 66,144 67,2031 68,537 69,1101 70,265 71,2296 72,190 73,2404 74,676 75,936 76,696 77,1943 78,203 79,2867 80,503 81,1623 82,912 83,3179 84,150 85,2275 86,999 87,1865 88,911 89,3750 90,110 91,2865 92,1121 93,2182 94,1285 95,3009 96,456 97,4603 98,1012 99,2249 100,901 101,4994 102,420 103,5158 104,1347 105,1500 106,1635 107,5562 108,539 109,5725 110,812 111,3123 112,1300 113,6178 114,502 115,4391 116,1852 117,3231 118,2048 119,5209 120,273 121,6478 122,2286 123,4081 124,2313 125,5810 126,536 127,8241 128,2568 [/CODE] The k for the case for prime with >=2 digits: [CODE] base,k 2,0 3,2 4,2 5,10 6,2 7,25 8,14 9,30 10,15 11,75 12,15 13,111 14,40 15,70 16,72 17,202 18,43 19,260 20,82 21,163 22,126 23,394 24,88 25,375 26,187 27,348 28,196 29,648 30,88 31,749 32,335 33,470 34,348 35,627 36,221 37,1089 38,450 39,684 40,385 41,1350 42,231 43,1495 44,579 45,764 46,685 47,1802 48,425 49,1674 50,628 51,1237 52,846 53,2311 54,549 55,1742 56,891 57,1575 58,1138 59,2894 60,458 61,3099 62,1316 63,1701 64,1470 65,2512 66,724 67,3766 68,1539 69,2370 70,1021 71,4245 72,1034 73,4500 74,1927 75,2242 76,1964 77,3802 78,1076 79,5295 80,1716 81,3495 82,2395 83,5861 84,1109 85,4476 86,2654 87,3879 88,2521 89,6768 90,1142 91,5466 92,2970 93,4467 94,3202 95,5671 96,1922 97,8078 98,2914 99,4697 100,2756 101,8774 102,1984 103,9137 104,3656 105,3683 106,4130 107,9883 108,2480 109,10270 110,2942 111,6478 112,3859 113,11051 114,2551 115,8490 116,4876 117,6765 118,5170 119,9691 120,2152 121,11515 122,5547 123,8024 124,5614 125,10609 126,2682 127,14030 128,6259 [/CODE][/QUOTE] This is why base 34 is harder than base 17, base 38 is harder than base 19, but base 42 is easier than base 21 [CODE] base number of unsolved families when searched to 10000 digits 17 2 34 33 19 5 38 77 21 3 42 0 (the largest prime has only 487 digits) [/CODE] |
[QUOTE=sweety439;562748]There are about exp(gamma*k) minimal primes in base n, where k = number of 2-digit numbers [I]xy[/I] in base n such that none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1
where exp(x) = e^x (e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] (2.718281828...), gamma is [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant"]Euler–Mascheroni constant[/URL] (0.5772156649...)) Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), where k = number of 2-digit numbers [I]xy[/I] in base n such that [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1[/QUOTE] exp(gamma*k) is the excepted value of the number of minimal primes base n, also the except value of the length of the largest minimal prime base n (when written in base n) |
[URL="https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf"]https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf[/URL]
|
[QUOTE=sweety439;562747]These bases are the bases <= 1024 which is not perfect odd power (of the form m^r with odd r>1) whose "minimal prime program" have GFN or half GFN remain, for the bases <= 1024 which is perfect odd power (of the form m^r with odd r>1):
* Cubes: ** Base 8: GFN in base 2 are either 2{0}1 or 4{0}1 in base 8, however, 2 and 401 are primes, thus, base 8 does not have GFN or half GFN remain. ** Base 27: half GFN in base 3 are either 1{D}E or 4{D}E in base 27, however, D is prime, thus, base 27 does not have GFN or half GFN remain. ** Base 64: GFN in base 2 are either 4{0}1 or G{0}1 in base 64, however, 41 and G01 are primes, thus, base 64 does not have GFN or half GFN remain. ** Base 125: half GFN in base 5 are either 2:{62}:63 or 12:{62}:63 in base 125, however, 2 is prime, but the family 12:{62}:63 does not have any known (probable) prime (the only known half GFN (probable) primes in base 5 are 3, 13, 2:63), thus, [B][I]base 125 has half GFN remain.[/I][/B] ** Base 216: GFN in base 6 are either 6:{0}:1 or 36:{0}:1 in base 216, however, 6:1 is prime, but the family 36:{0}:1 does not have any known prime (the only known GFN primes in base 6 are 7, 37, 6:1), thus, [B][I]base 216 has GFN remain.[/I][/B] ** Base 343: half GFN in base 7 are either 3:{171}:172 or 24:{171}:172 in base 343, however, 3 is prime, but the family 24:{171}:172 does not have any known (probable) prime (the only known half GFN (probable) prime in base 7 is 3:172), thus, [B][I]base 343 has half GFN remain.[/I][/B] ** Base 512: GFN in base 2 are 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 128:{0}:1, or 256:{0}:1 in base 512, however, 2 and 128:1 are primes, but the families 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 128:1), thus, [B][I]base 512 has GFN remain.[/I][/B] ** Base 729: half GFN in base 3 are either 4:{364}:365 or 40:{364}:365 in base 729, however, 40:364:365 and 4:364:364:364:364:365 are primes, thus, base 729 does not have GFN or half GFN remain. ** Base 1000: GFN in base 10 are either 10:{0}:1 or 100:{0}:1 in base 1000, and both families do not have any known prime (the only known GFN primes in base 10 are 11 and 101), thus, [B][I]base 1000 has GFN remain.[/I][/B] * 5th powers: ** Base 32: GFN in base 2 are 2{0}1, 4{0}1, 8{0}1, or G{0}1 in base 32, however, 2 and 81 are primes, but the families 4{0}1 and G{0}1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, H, 81, 2001), thus, [B][I]base 32 has GFN remain.[/I][/B] ** Base 243: half GFN in base 3 are 1:{121}:122, 4:{121}:122, 13:{121}:122, or 40:{121}:122 in base 243, however, 1:121:121:122, 4:121, 13, 40:121:121:121:121:121:121:121:121:121:121:121:122 are primes, thus, base 243 does not have GFN or half GFN remain. ** Base 1024: GFN in base 2 are 4:{0}:1, 16:{0}:1, 64:{0}:1, or 256:{0}:1 in base 1024, however, 64:1 is prime, but the families 4:{0}:1, 16:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 64:1), thus, [B][I]base 1024 has GFN remain.[/I][/B] * 7th powers: ** Base 128: GFN in base 2 are 2:{0}:1, 4:{0}:1, or 16:{0}:1 in base 128, however, 2 and 4:0:1 are primes, but the family 16:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 2:1, 4:0:1), thus, [B][I]base 128 has GFN remain.[/I][/B][/QUOTE] The smallest generalized repunit prime in base b (if exists) is [I]always[/I] minimal prime in base b, since it is 111...111 in base b Thus, a given base b which is not perfect power (of the form m^r with r>1) whose "minimal prime program" have generalized repunit prime remain if and only if there are no known generalized repunit prime in base b, such bases <= 1024 (perfect powers excluded) are 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 For the bases <= 1024 which is perfect power (of the form m^r with r>1): * Squares: ** Base 4: GRU in base 2 are 1{3} in base 4, however, 3 is prime, thus, base 4 does not have GRU remain. ** Base 9: GRU in base 3 are 1{4} in base 9, however, 14 is prime, thus, base 9 does not have GRU remain. ** Base 16: GRU in base 2 are either 1{F} or 7{F} in base 16, however, 1F and 7 are primes, thus, base 16 does not have GRU remain. ** Base 25: GRU in base 5 are 1{6} in base 25, however, 16 is prime, thus, base 25 does not have GRU remain. ** Base 36: GRU in base 6 are 1{7} in base 36, however, 7 is prime, thus, base 36 does not have GRU remain. ** Base 49: GRU in base 7 are 1:{8} in base 49, however, 1:8:8 is prime, thus, base 49 does not have GRU remain. ** Base 64: GRU in base 2 are either 1:{63} or 31:{63} in base 64, however, 1:63 and 31 are primes, thus, base 64 does not have GRU remain. ** Base 81: GRU in base 3 are either 1:{40} or 13:{40} in base 81, however, 1:40:40:40 and 13 are primes, thus, base 81 does not have GRU remain. ** Base 100: GRU in base 10 are 1:{11} in base 100, however, 11 is prime, thus, base 100 does not have GRU remain. ** Base 121: GRU in base 11 are 1:{12} in base 121, however, 1:12:12:12:12:12:12:12:12 is prime, thus, base 121 does not have GRU remain. ** Base 144: GRU in base 12 are 1:{13} in base 144, however, 13 is prime, thus, base 144 does not have GRU remain. ... ** Base 1024: GRU in base 2 are 1:{1023}, 7:{1023}, 127:{1023}, or 511:{1023} in base 1024, however, 1:1023:1023:1023, 7, 127, 511:1023 are primes, thus, base 1024 does not have GRU remain. * Cubes: ** Base 8: GRU in base 2 are either 1{7} or 3{7} in base 8, however, 7 is prime, thus, base 8 does not have GRU remain. ** Base 27: GRU in base 3 are either 1{D} or 4{D} in base 27, however, D is prime, thus, base 27 does not have GRU remain. ** Base 64: GRU in base 2 are either 1:{63} or 31:{63} in base 64, however, 1:63 and 31 are primes, thus, base 64 does not have GRU remain. ** Base 125: GRU in base 5 are either 1:{31} or 6:{31} in base 125, however, 31 is prime, thus, base 125 does not have GRU remain. ** Base 216: GRU in base 6 are either 1:{43} or 7:{43} in base 216, however, 43 is prime, thus, base 216 does not have GRU remain. ** Base 343: GRU in base 7 are either 1:{57} or 8:{57} in base 343, however, 1:57:57:57:57 and 8:57 are primes, thus, base 343 does not have GRU remain. ** Base 512: GRU in base 2 are 1:{511}, 3:{511}, 15:{511}, 31:{511}, 127:{511}, or 255:{511} in base 512, however, 1:511:511, 3, 15:511, 31, 127, 255:511 are primes, thus, base 512 does not have GRU remain. ** Base 729: GRU in base 3 are either 1:{364} or 121:{364} in base 729, however, 1:364 and 121:364:364:364:364:364:364:364:364:364:364:364 are primes, thus, base 729 does not have GRU remain. ** Base 1000: GRU in base 10 are either 1:{111} or 11:{111} in base 1000, however, 1:111:111:111:111:111:111 and 11 are primes, thus, base 1000 does not have GRU remain. * 5th powers: ** Base 32: GRU in base 2 are 1{V}, 3{V}, 7{V}, or F{V} in base 32, however, V is prime, thus, base 32 does not have GRU remain. ** Base 243: GRU in base 3 are 1:{121}, 4:{121}, 13:{121}, or 40:{121} in base 243, however, 1:121:121:121:121:121:121:121:121:121:121:121:121:121:121, 4:121, 13 are primes, but the family 40:{121} does not have any known (probable) prime (there are no known numbers in [URL="https://oeis.org/A028491"]OEIS A028491[/URL] which is == 4 mod 5), thus, [B][I]base 243 has GRU remain.[/I][/B] ** Base 1024: GRU in base 2 are 1:{1023}, 7:{1023}, 127:{1023}, or 511:{1023} in base 1024, however, 1:1023:1023:1023, 7, 127, 511:1023 are primes, thus, base 1024 does not have GRU remain. * 7th powers: ** Base 128: GRU in base 2 are 1:{127}, 3:{127}, 7:{127}, 15:{127}, 31:{127}, or 63:{127} in base 128, however, 127 is prime, thus, base 128 does not have GRU remain. |
In [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski problem[/URL] base b, the prime for a k-value <b is "minimal prime base b" if and only if k is not prime.
In [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel problem[/URL] base b, the prime for a k-value <b is "minimal prime base b" if and only if neither k-1 nor b-1 is prime. However, if we exclude the single-digit primes from the set (i.e. the minimal string of the set of prime numbers >= b in base b, see problem [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), then the prime for Sierpinski/Riesel problems base b for a k-value <b is always "minimal prime base b", this is why the "minimal prime problem" for the prime numbers >= b in base b is more interesting, since single-digit primes are trivial, like that in Sierpinski/Riesel problems base b, n=0 is trivial, since the corresponding number is just k+1 or k-1, and thus CRUS requires n>=1, and of course the CRUS Sierpinski/Riesel problems (requiring n>=1) is much harder than the same problem which n=0 is allowed, similarly, finding the minimal set of the strings for primes in base b with at least two digits in base b is much harder than finding the minimal set of the strings for primes (including the single-digit primes in base b) in base b, e.g. * In base 7, the largest minimal prime is 11111, but if single-digit primes are excluded, then a much-larger prime 33333333333333331 is minimal prime. * In base 8, the largest minimal prime is 444444441, but if single-digit primes are excluded, then a much-larger prime 7777777777771 is minimal prime. * In base 10, the largest minimal prime is 66600049, but if single-digit primes are excluded, then a much-larger prime 555555555551 is minimal prime. * In base 14, the largest minimal prime is 40[SUB]83[/SUB]49, but if single-digit primes are excluded, then a much-larger prime 4D[SUB]19698[/SUB] is minimal prime. * In base 17, there are only 2 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 74[SUB]4904[/SUB] is minimal prime. * In base 21, there are only 3 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 5D0[SUB]19848[/SUB]1 is minimal prime. * In base 30, the largest minimal prime is C0[SUB]1022[/SUB]1, but if single-digit primes are excluded, then a much-larger prime OT[SUB]34205[/SUB] is minimal prime. * In base 32, there are 78 unsolved families when searched to length 10000, but if single-digit primes are excluded, then the unsolved family S{V} is searched up to length 2000001 by CRUS with no prime found. * In base 33, there are 33 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 130[SUB]23614[/SUB]1 is minimal prime. * In base 35, there are only 15 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 1B0[SUB]56061[/SUB]1 is minimal prime. * In base 37, if single-digit primes are excluded, then the unsolved family 2K{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family {I}J is searched up to length 1048575 by GFN search with no (probable) prime found. * In base 38, if single-digit primes are excluded, then there are four large known minimal primes 20[SUB]2728[/SUB]1, V0[SUB]1527[/SUB]1, Lb[SUB]1579[/SUB], ab[SUB]136211[/SUB], also the unsolved family 1{0}V is searched up to length 185001 by Peter Košinár with no (probable) prime found. * In base 42, the largest minimal prime is R[SUB]486[/SUB]1, but if single-digit primes are excluded, then a much-larger prime 2f[SUB]2523[/SUB] is minimal prime. * In base 43, if single-digit primes are excluded, then the unsolved family 3b{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family 2{7} is searched up to length 50001 by Dylan Delgado with no (probable) prime found. * In base 48, if single-digit primes are excluded, then there is a large known minimal prime T0[SUB]133041[/SUB]1. * In base 60, if single-digit primes are excluded, then the unsolved family Z{x} is searched up to length 100001 by CRUS with no prime found. References: [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub[/URL] (Base 17 7{4} family) [URL="https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31"]https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31[/URL] (Base 38 1{0}V family) [URL="https://github.com/curtisbright/mepn-data"]https://github.com/curtisbright/mepn-data[/URL] (original minimal prime problem, bases 2 to 30) [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] (original minimal prime problem, bases 28 to 50) |
[QUOTE=sweety439;560430]The "minimal prime problem" is solved only in bases 2~16, 18, 20, 22~24, 30, 42, and maybe 60
[CODE] b, length of largest minimal prime base b, number of minimal primes base b 2, 2, 2 3, 3, 3 4, 2, 3 5, 5, 8 6, 5, 7 7, 5, 9 8, 9, 15 9, 4, 12 10, 8, 26 11, 45, 152 12, 8, 17 13, 32021, 228 14, 86, 240 15, 107, 100 16, 3545, 483 18, 33, 50 20, 449, 651 22, 764, 1242 23, 800874, 6021 24, 100, 306 30, 1024, 220 42, 4551, 487 60, ?, ? (in theory, <2000 digits) [/CODE][/QUOTE] The lower bound of the largest minimal prime in base 60 is e[SUB]1937[/SUB]1, also Q[SUB]896[/SUB]1 is minimal prime in base 60 The values of them are 77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 and 44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161 |
For the minimal problem in base 60, I checked these families: (x is any given digit)
* x:{0}:1 * x:{0}:49 (x not divisible by 7, x != 10) * {x}:1 * {x}:49 (x not divisible by 7) * x:{49} (x not divisible by 7) All these families have known proven primes, except {40}:1, which only has known strong PRP * x:{14}:49 (x not divisible by 7) * x:{21}:49 (x not divisible by 7) * x:{28}:49 (x not divisible by 7) * x:{35}:49 (x not divisible by 7) * x:{42}:49 (x not divisible by 7) Only 46:{42}:49 family has no known primes, even no strong PRPs in this family are known * {14}:x:49 (x not divisible by 7) * {21}:x:49 (x not divisible by 7) * {28}:x:49 (x not divisible by 7) * {35}:x:49 (x not divisible by 7) * {42}:x:49 (x not divisible by 7) All these families have known proven primes, except {42}:30:49, which only has known strong PRP |
[QUOTE=sweety439;564341]For the minimal problem in base 60, I checked these families: (x is any given digit)
* x:{0}:1 * x:{0}:49 (x not divisible by 7, x != 10) * {x}:1 * {x}:49 (x not divisible by 7) * x:{49} (x not divisible by 7) All these families have known proven primes, except {40}:1, which only has known strong PRP * x:{14}:49 (x not divisible by 7) * x:{21}:49 (x not divisible by 7) * x:{28}:49 (x not divisible by 7) * x:{35}:49 (x not divisible by 7) * x:{42}:49 (x not divisible by 7) Only 46:{42}:49 family has no known primes, even no strong PRPs in this family are known * {14}:x:49 (x not divisible by 7) * {21}:x:49 (x not divisible by 7) * {28}:x:49 (x not divisible by 7) * {35}:x:49 (x not divisible by 7) * {42}:x:49 (x not divisible by 7) All these families have known proven primes, except {42}:30:49, which only has known strong PRP[/QUOTE] 46:{42}:49 family has trivial factor of 53, thus not need to be checked. Also checked these families: * 10:10:{0}:49 * 10:x:{0}:49 (x = 14, 21, 28, 35, 42, 49) * x:10:{0}:49 (x = 14, 21, 28, 35, 42, 49) |
Smallest prime in given simple family in base 60:
{x}:1 [CODE] 1,61 2,7321 3,181 4,241 5,18301 6,21961 7,421 8,379574237281 9,541 10,601 11,661 12,208167532693722559227661016949152542372881355921 13,47581 14,51241 15,3294901 16,45548908474561 17,1021 18,65881 19,15024813541 20,1201 21,276772861 22,1321 23,1381 24,316311841 25,5491501 26,44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161 27,1621 28,102481 29,1741 30,1801 31,1861 32,93237738291439869343927223105084745762711864406779661016949121 33,1565743728781 34,1613190508441 35,7688101 36,2161 37,2221 38,2281 39,2341 40,77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 41,150061 42,2521 43,122412691525381 44,9665041 45,164701 46,606264361 47,172021 48,37957423681 49,10763341 50,3001 51,3061 52,3121 53,3181 54,197641 55,3301 56,3361 57,28550279937155564567041751967467111314451825663380318242711864406779661016949152542372881355932203389830508474576271186440621 58,212281 59,3541 [/CODE] {x}:49 [CODE] 1,109 2,7369 3,229 4,52718689 5,349 6,409 8,105437329 9,12329714402004875501336027803574282441328813559322033898305084745762711864406779661016989 10,131796649 11,709 12,769 13,829 15,54949 16,1009 17,1069 18,1129 19,250413589 20,1249 22,811681549016949152569 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,109849 31,408569509 32,119904498831584194115132745762711864406779661016949169 33,2029 34,2089 36,3028642546542661941210719954058267766581114796637317680350572474576271186440677966101694915254237288135593220338983050847457627118644067809 37,2269 38,6490719457627129 39,2389 40,146449 41,540366109 43,157429 44,2689 45,2749 46,168409 47,172069 48,6375389621369491525423729 50,3049 51,3109 52,3169 53,3229 54,197689 55,9394462372881349 57,3469 58,3529 59,46655999989 [/CODE] x:{49} [CODE] 1,109 2,611389 3,229 4,17389 5,349 6,409 8,411996203389 9,77035957924881355932203389 10,14590162042372436009914299567562900888905762711864406779661016949152542372881355932203389 11,709 12,769 13,829 15,56989 16,1009 17,1069 18,1129 19,71389 20,1249 22,82189 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,110989 31,6875389 32,118189 33,2029 34,2089 36,132589 37,2269 38,2384559087006591915952856949152542372881355932203389 39,2389 40,146989 41,150589 43,4451424541432606372881355932203389 44,2689 45,2749 46,606923389 47,133894812203389 48,10547389 50,3049 51,3109 52,3169 53,3229 54,197389 55,200989 57,3469 58,3529 59,215389 [/CODE] x:{0}:1 [CODE] 1,61 2,432001 3,181 4,241 5,2406149016991872132210992275783680000000000000000000000000000000000000000001 6,21601 7,421 8,62691331276800000000000001 9,541 10,601 11,661 12,43201 13,168480001 14,10886400001 15,54001 16,57601 17,1021 18,13996800001 19,20370649768045944216583302410995908998024578122182598718037950464000000000000000000000000000000000000000000000000000000000000000000000000000000001 20,1201 21,3527193600000001 22,1321 23,1381 24,5184001 25,90001 26,93601 27,1621 28,100801 29,1741 30,1801 31,1861 32,115201 33,118801 34,122401 35,126001 36,2161 37,2221 38,2281 39,2341 40,8640001 41,531360001 42,2521 43,2006208000001 44,965225828176625664000000000000000000001 45,125971200000001 46,165601 47,2192832000001 48,172801 49,176401 50,3001 51,3061 52,3121 53,3181 54,93551073780643988500363379682469478400000000000000000000000000000000000000000001 55,3301 56,3361 57,205201 58,(trivial factor of 59) 59,3541 [/CODE] x:{0}:49 [CODE] 1,109 2,93312000049 3,229 4,14449 5,349 6,409 8,6220800049 9,1944049 10,(trivial factor of 59) 11,709 12,769 13,829 15,54049 16,1009 17,1069 18,1129 19,68449 20,1249 22,61585920000049 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,6480049 31,6696049 32,115249 33,2029 34,2089 36,466560049 37,2269 38,136849 39,2389 40,1866240000049 41,1912896000049 43,154849 44,2689 45,2749 46,2146176000049 47,169249 48,172849 50,3049 51,3109 52,3169 53,3229 54,699840049 55,1625362428001224684562289212884750459919231127678405836800000000000000000000000000000000000000000000000000000000000000000000049 57,3469 58,3529 59,12744049 [/CODE] |
x:{14}:49
[CODE] 1,109 2,8089 3,229 4,15289 5,349 6,409 8,1779289 9,33289 10,132675289 11,709 12,769 13,829 15,20055578263423416146440677966101694915289 16,1009 17,1069 18,1129 19,249315289 20,1249 22,1037502915289 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,6531289 31,6747289 32,116089 33,2029 34,2089 36,130489 37,2269 38,495555289 39,2389 40,144889 41,103182884390444449640082580739273165590541602171762902235022709553359196234458364272060077833460415740804281702234241751684562368942580406237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915289 43,155689 44,2689 45,2749 46,465965333694915289 47,10203289 48,37509315289 50,3049 51,3109 52,3169 53,3229 54,702915289 55,11931289 57,3469 58,3529 59,213289 [/CODE] x:{21}:49 [CODE] 1,109 2,508909 3,229 4,56452909 5,349 6,409 8,30109 9,94286240542372909 10,37309 11,709 12,769 13,829 15,42986782372909 16,1009 17,1069 18,1129 19,69709 20,1249 22,4025817357839737065999049857469767607759032979106716427631968228584180718644067796610169491525423728813559322033898305084745762711864406779661016949152542372909 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,66078257013152542372909 31,112909 32,6988909 33,2029 34,2089 36,101773342372909 37,2269 38,497092909 39,2389 40,31380772909 41,535972909 43,156109 44,2689 45,2749 46,166909 47,170509 48,10444909 50,3049 51,3109 52,3169 53,3229 54,195709 55,717412909 57,3469 58,3529 59,12820909 [/CODE] x:{28}:49 [CODE] 1,109 2,8929 3,229 4,57990529 5,349 6,409 8,30529 9,34129 10,2262529 11,709 12,769 13,829 15,3342529 16,1009 17,1069 18,1129 19,4206529 20,1249 22,80929 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,394950529 31,113329 32,116929 33,2029 34,2089 36,472710529 37,2269 38,8310529 39,2389 40,8742529 41,4212156168942609355932203389830529 43,9390529 44,2689 45,2749 46,167329 47,1041447436333870756881355932203389830529 48,13781461655186262022942372881355932203389830529 50,3049 51,3109 52,3169 53,3229 54,331946136337921041355932203389830529 55,199729 57,3469 58,3529 59,214129 [/CODE] x:{35}:49 [CODE] 1,109 2,9349 3,229 4,771484637288149 5,349 6,409 8,30949 9,34549 10,38149 11,709 12,769 13,829 15,56149 16,1009 17,1069 18,1129 19,70549 20,1249 22,81349 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,6608149 31,113749 32,25344488149 33,2029 34,2089 36,131749 37,2269 38,23335844534237288149 39,2389 40,526088149 41,149749 43,33898088149 44,2689 45,2749 46,10064149 47,185120325561205169675135454687984165581220276782105983361193164168670838265752820260904448577571678108045870876576322502449572547744055070802594515527926156691477915758601548938853501303980626387044701079429027352148995318212751455646534728345092572787387347410490834215458172781888350653835992601161966471095319325576754886993561238893441826145675686736477047030489460058140333352185261905102459753937730302319495540099005728377120233422356413887591971257315898743429492383226804915072909057608763313496683212160248130680520291356694770983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288149 48,10496149 50,3049 51,3109 52,3169 53,3229 54,196549 55,12008149 57,3469 58,3529 59,12872149 [/CODE] x:{42}:49 [CODE] 1,109 2,9769 3,229 4,170945053505084745769 5,349 6,409 8,1881769 9,2097769 10,38569 11,709 12,769 13,829 15,56569 16,1009 17,1069 18,1129 19,70969 20,1249 22,81769 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,110569 31,6849769 32,7065769 33,2029 34,2089 36,132169 37,2269 38,139369 39,2389 40,31657545769 41,150169 43,33990345769 44,2689 45,2749 46,(trivial factor of 53) 47,10305769 48,1767252099905084745769 50,3049 51,3109 52,3169 53,3229 54,11817769 55,200569 57,3469 58,3529 59,12897769 [/CODE] {14}:x:49 [CODE] 1,3074509 2,31719478566747843550804041300349830508474576271186440677966101694914569 3,11070914629 4,11070914689 5,184514749 6,11070914809 8,50929 9,50989 10,3075049 11,51109 12,51169 13,51229 15,51349 16,3075409 17,8608743701694915469 18,516524622101694915529 19,11070915589 20,184515649 22,51769 23,51829 24,664254915889 25,51949 26,52009 27,52069 29,52189 30,52249 31,39855294916309 32,52369 33,3076429 34,52489 36,52609 37,664254916669 38,30991477326101694916729 39,3076789 40,11070916849 41,664254916909 43,184517029 44,53089 45,53149 46,53690481146394350663097813232332427347712798372881355932203389830508474576271186440677966101694917209 47,53269 48,664254917329 50,86756301967596040677966101694917449 51,11070917509 52,53569 53,53629 54,3077689 55,3147493094329085095522234576271186440677966101694917749 57,30991477326101694917869 58,184517929 59,39855294917989 [/CODE] {21}:x:49 [CODE] 1,75709 2,10041238653656949152542371769 3,4611829 4,276771889 5,16606371949 6,4612009 8,76129 9,616626663337578078627630562878800705084745762711864406779661016949152542372189 10,76249 11,645936492161087054050399981653446997706215532953486454915466633416169490161247457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372309 12,76369 13,996382372429 15,4612549 16,4612609 17,4612669 18,276772729 19,4612789 20,4612849 22,4612969 23,77029 24,276773089 25,16606373149 26,4613209 27,77269 29,276773389 30,4613449 31,77509 32,77569 33,4613629 34,77689 36,16606373809 37,276773869 38,77929 39,16606373989 40,78049 41,4614109 43,78229 44,59782942374289 45,4614349 46,6071553036900241310806779661016949152542374409 47,276774469 48,61187265753757414256952240162711864406779661016949152542374529 50,78649 51,774786933152542374709 52,996382374769 53,4614829 54,78889 55,52209837304507200077039765027430616624964848906126524755206192493399488812789496523127382219681677062442951316647311186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542374949 57,4615069 58,996382375129 59,276775189 [/CODE] {28}:x:49 [CODE] 1,81583021005009885675936320216949152542372881355932203389828909 2,85460868854823834608016221386772928746969615932515587546395079909292691525423728813559322033898305084745762711864406779661016949152542372881355932203389828969 3,22141829029 4,101089 5,101149 6,101209 8,6149329 9,6149389 10,101449 11,286958123389829509 12,10410756236111524881355932203389829569 13,369029629 15,101749 16,79710589829809 17,101869 18,101929 19,1328509829989 20,6150049 22,13388318204875932203389830169 23,102229 24,6150289 25,79710589830349 26,102409 27,1328509830469 29,6150589 30,6150649 31,29828045081330194812832118462406904082062665762711864406779661016949152542372881355932203389830709 32,102769 33,102829 34,286958123389830889 36,41126317117261134141412615065804034201951786857394034478949601447152874939439946846072866224064662010793220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389831009 37,103069 38,6151129 39,1328509831189 40,369031249 41,369031309 43,286958123389831429 44,1328509831489 45,103549 46,1033049244203389831609 47,103669 48,6151729 50,1328509831849 51,286958123389831909 52,103969 53,37478722450001489572881355932203389832029 54,104089 55,104149 57,6152269 58,1328509832329 59,6152389 [/CODE] {35}:x:49 [CODE] 1,81475573154959817379255597463815552667176545024717505666213665960638224320021112459781503325113716943629003292963749951159320447728997238620350548107901686289125472897903137985503182233361374542375162900682265823294796999737443473337843435573308433456959404707058320810457306411455933006101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237286109 2,1660637286169 3,126229 4,17128518426043835517434182302188908474576271186440677966101694915254237286289 5,126349 6,36429318221401447864840677966101694915254237286409 8,77478693315254237286529 9,7686589 10,7686649 11,27677286709 12,99638237286769 13,461286829 15,126949 16,1660637287009 17,461287069 18,1660637287129 19,127189 20,127249 22,607155303690024131080677966101694915254237287369 23,131145545597045212313426440677966101694915254237287429 24,36429318221401447864840677966101694915254237287489 25,127549 26,127609 27,127669 29,27677287789 30,127849 31,358697654237287909 32,461287969 33,21521859254237288029 34,99638237288089 36,7688209 37,1660637288269 38,1660637288329 39,128389 40,128449 41,128509 43,128629 44,461288689 45,128749 46,7688809 47,358697654237288869 48,7688929 50,129049 51,7689109 52,129169 53,129229 54,129289 55,21521859254237289349 57,129469 58,129529 59,129589 [/CODE] {42}:x:49 [CODE] 1,9223309 2,293439587902707111288657562346647653394212382542384936694711075914089081641468094666645578615570550709924319586579296659981779875032195661701462202096102293874185480241052744144833349365537276799785969205297493379562404092952314178395004440542139367761173708310416661107311234222306240316209631923293822273260390016005704126405409242393725556515881932954120677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084743369 3,151429 4,260268905902788122033898305084743489 5,151549 6,151609 8,151729 9,95158435700243530652412123901049491525423728813559322033898305084743789 10,151849 11,151909 12,151969 13,152029 15,334707955121898305084744149 16,33212744209 17,9224269 18,25826231105084744329 19,152389 20,33212744449 22,1992764744569 23,152629 24,9224689 25,33212744749 26,152809 27,553544869 29,152989 30,1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049 31,3452320032561365140374087785000799083572067796610169491525423728813559322033898305084745109 32,33212745169 33,1992764745229 34,9225289 36,153409 37,153469 38,153529 39,153589 40,153649 41,9225709 43,202385101230008043693559322033898305084745829 44,153889 45,153949 46,430437185084746009 47,9226069 48,1992764746129 50,553546249 51,33212746309 52,154369 53,1204948638438833898305084746429 54,95158435700243530652412123901049491525423728813559322033898305084746489 55,9226549 57,154669 58,33212746729 59,154789 [/CODE] |
{49}:x:49
[CODE] 1,176509 2,10760569 3,176629 4,645800689 5,1807836177355932200749 6,176809 8,8369611932200929 9,176989 10,5179663972035655860472096728181925922711864406779661016949152542372881355932201049 11,177109 12,38748201169 13,645801229 15,3370877628514757222772400494683673158596409477797247719358711849483581786873706143158543146664730075705489224200224093772041630182187652701699645946277442899767500755812009374102776457086216749247209626019900398116606709317609816539188162397141065837083072359107434141070736000398492419524277805139751989637641458276851772548616402672230694356610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932201349 16,177409 17,10761469 18,10761529 19,177589 20,10761649 22,38748201769 23,645801829 24,177889 25,177949 26,645802009 27,178069 29,30130602955932202189 30,178249 31,645802309 32,84346404690718372881355932202369 33,10762429 34,178489 36,178609 37,1093129404791710112542372881355932202669 38,10762729 39,10762789 40,390492614308881355932202849 41,178909 43,179029 44,179089 45,38748203149 46,179209 47,179269 48,409053718397842884147674224095289021756765923247261869559322033898305084745762711864406779661016949152542372881355932203329 50,645803449 51,38748203509 52,10763569 53,10763629 54,179689 55,179749 57,10763869 58,1807836177355932203929 59,179989 [/CODE] 10:x:{0}:49 [CODE] 614,477446400049 621,37309 628,8138880049 635,38149 642,38569 649,140184049 [/CODE] x:10:{0}:49 [CODE] 850,183600049 1270,76249 1690,101449 2110,3571525142839296000000000000000049 2530,151849 2950,38232000049 [/CODE] |
Other possible such simple families:
10:10:{0}:49 prime is 2196049 58:58:{0}:1 prime is 212281 10:{0}:49:49 prime is 6046617600000002989 10:{0}:10:49 prime is 466560000649 58:{0}:58:1 prime is 212281 |
14:{0}:x:49
[CODE] 1,24253982091278071092686802139899494400000000000000000000000000000000000000000109 2,1009 3,1069 4,1129 5,3024349 6,1249 8,50929 9,1429 10,1489 11,1549 12,1609 13,1669 15,1789 16,5361922889755312850308759177923802244504459923744908273254400000000000000000000000000000000000000000000000000000000000000000000001009 17,3025069 18,3025129 19,2029 20,2089 22,51769 23,2269 24,10886401489 25,2389 26,52009 27,52069 29,52189 30,2689 31,2749 32,52369 33,181442029 34,52489 36,3049 37,3109 38,3169 39,3229 40,39191040002449 41,3026509 43,3469 44,3529 45,53149 46,181442809 47,3709 48,3769 50,3889 51,181443109 52,53569 53,53629 54,4129 55,(trivial factor of 59) 57,3027469 58,307117308965289984000000000000000003529 59,10886403589 [/CODE] 21:{0}:x:49 [CODE] 1,75709 2,1429 3,1489 4,1549 5,1609 6,1669 8,1789 9,211631616000000589 10,76249 11,272160709 12,2029 13,2089 15,16329600949 16,2269 17,4537069 18,2389 19,4537189 20,272161249 22,12697896960000001369 23,2689 24,2749 25,2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,3527193600001609 27,77269 29,3049 30,3109 31,3169 32,3229 33,272162029 34,77689 36,3469 37,3529 38,77929 39,16329602389 40,3709 41,3769 43,3889 44,272162689 45,4538749 46,278553138848124030953717760000000000000000000000000002809 47,4129 48,(trivial factor of 59) 50,78649 51,12697896960000003109 52,4539169 53,979776003229 54,4549 55,58786560003349 57,4729 58,4789 59,592433080565760000000000003589 [/CODE] 28:{0}:x:49 [CODE] 1,1789 2,789910774087680000000000000169 3,21772800229 4,101089 5,2029 6,2089 8,21772800529 9,2269 10,101449 11,2389 12,6048769 13,6048829 15,101749 16,2689 17,2749 18,101929 19,47394646445260800000000000001189 20,362881249 22,3049 23,3109 24,3169 25,3229 26,102409 27,78382080001669 29,3469 30,3529 31,80223303988259720914670714880000000000000000000000000000001909 32,102769 33,3709 34,3769 36,3889 37,103069 38,282175488000002329 39,6050389 40,4129 41,(trivial factor of 59) 43,1306368002629 44,2316350688374295151333383964863082569625926687057800374045900800000000000000000000000000000000000000000000000000000000000000000000000002689 45,103549 46,362882809 47,4549 48,21772802929 50,4729 51,4789 52,103969 53,4909 54,4969 55,104149 57,1039694019687845983054132464844800000000000000000000000000000000003469 58,5209 59,78382080003589 [/CODE] 35:{0}:x:49 [CODE] 1,7560109 2,2269 3,126229 4,2389 5,126349 6,16456474460160000000000000409 8,453600529 9,2689 10,2749 11,5878656000000709 12,76187381760000000000769 13,97977600000829 15,3049 16,3109 17,3169 18,3229 19,127189 20,127249 22,3469 23,3529 24,27855313884812403095371776000000000000000000000000000001489 25,127549 26,3709 27,3769 29,3889 30,127849 31,7561909 32,7561969 33,4129 34,(trivial factor of 59) 36,453602209 37,1632960002269 38,7562329 39,128389 40,4549 41,128509 43,4729 44,4789 45,128749 46,4909 47,4969 48,352719360000002929 50,129049 51,5209 52,129169 53,129229 54,129289 55,5449 57,5569 58,129529 59,5689 [/CODE] 42:{0}:x:49 [CODE] 1,9072109 2,2689 3,2749 4,32659200289 5,151549 6,151609 8,3049 9,3109 10,3169 11,3229 12,151969 13,152029 15,3469 16,3529 17,117573120001069 18,5614347706314368308492315310161920000000000000000000000000000000000001129 19,3709 20,3769 22,3889 23,152629 24,9073489 25,16850366041100048016273224228350264013299690448515611349014072815152914037687200443877077133470937907200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,4129 27,(trivial factor of 59) 29,152989 30,5485491486720000000001849 31,32659201909 32,544321969 33,4549 34,9074089 36,4729 37,4789 38,153529 39,4909 40,4969 41,117573120002509 43,20686430901833768712610953618533187671699305261361528832000000000000000000000000000000000000000000000000000000000000000002629 44,5209 45,153949 46,544322809 47,544322869 48,5449 50,5569 51,387215843042271258003015621585831529981559389785592810438298620409934449687415027238102244809934838667668586191368584982641890866197562403890778944989683725107200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003109 52,5689 53,5749 54,32659203289 55,5869 57,154669 58,1959552003529 59,154789 [/CODE] 49:{0}:x:49 [CODE] 1,3049 2,3109 3,3169 4,3229 5,3379480093071544116029035238523182050306352376786936240751182230865067894959768623789993478122963368981956666059054838206787747840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349 6,176809 8,3469 9,3529 10,10584649 11,177109 12,3709 13,3769 15,3889 16,177409 17,137168640001069 18,140390781979454511600673751040000000000000000000000000000001129 19,4129 20,(trivial factor of 59) 22,635041369 23,38102401429 24,177889 25,177949 26,4549 27,178069 29,4729 30,4789 31,10585909 32,4909 33,4969 34,178489 36,178609 37,5209 38,635042329 39,1382343854653440000000000002389 40,10586449 41,5449 43,5569 44,179089 45,5689 46,5749 47,179269 48,5869 50,137168640003049 51,29628426240000003109 52,106662334464000000003169 53,635043229 54,6229 55,179749 57,10587469 58,6469 59,6529 [/CODE] |
[URL="https://raw.githubusercontent.com/xayahrainie4793/primes/master/kernel60.txt"]minimal primes in base 60 up to 2^32[/URL]
|
[QUOTE=sweety439;564438]14:{0}:x:49
[CODE] 1,24253982091278071092686802139899494400000000000000000000000000000000000000000109 2,1009 3,1069 4,1129 5,3024349 6,1249 8,50929 9,1429 10,1489 11,1549 12,1609 13,1669 15,1789 16,5361922889755312850308759177923802244504459923744908273254400000000000000000000000000000000000000000000000000000000000000000000001009 17,3025069 18,3025129 19,2029 20,2089 22,51769 23,2269 24,10886401489 25,2389 26,52009 27,52069 29,52189 30,2689 31,2749 32,52369 33,181442029 34,52489 36,3049 37,3109 38,3169 39,3229 40,39191040002449 41,3026509 43,3469 44,3529 45,53149 46,181442809 47,3709 48,3769 50,3889 51,181443109 52,53569 53,53629 54,4129 55,(trivial factor of 59) 57,3027469 58,307117308965289984000000000000000003529 59,10886403589 [/CODE] 21:{0}:x:49 [CODE] 1,75709 2,1429 3,1489 4,1549 5,1609 6,1669 8,1789 9,211631616000000589 10,76249 11,272160709 12,2029 13,2089 15,16329600949 16,2269 17,4537069 18,2389 19,4537189 20,272161249 22,12697896960000001369 23,2689 24,2749 25,2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,3527193600001609 27,77269 29,3049 30,3109 31,3169 32,3229 33,272162029 34,77689 36,3469 37,3529 38,77929 39,16329602389 40,3709 41,3769 43,3889 44,272162689 45,4538749 46,278553138848124030953717760000000000000000000000000002809 47,4129 48,(trivial factor of 59) 50,78649 51,12697896960000003109 52,4539169 53,979776003229 54,4549 55,58786560003349 57,4729 58,4789 59,592433080565760000000000003589 [/CODE] 28:{0}:x:49 [CODE] 1,1789 2,789910774087680000000000000169 3,21772800229 4,101089 5,2029 6,2089 8,21772800529 9,2269 10,101449 11,2389 12,6048769 13,6048829 15,101749 16,2689 17,2749 18,101929 19,47394646445260800000000000001189 20,362881249 22,3049 23,3109 24,3169 25,3229 26,102409 27,78382080001669 29,3469 30,3529 31,80223303988259720914670714880000000000000000000000000000001909 32,102769 33,3709 34,3769 36,3889 37,103069 38,282175488000002329 39,6050389 40,4129 41,(trivial factor of 59) 43,1306368002629 44,2316350688374295151333383964863082569625926687057800374045900800000000000000000000000000000000000000000000000000000000000000000000000002689 45,103549 46,362882809 47,4549 48,21772802929 50,4729 51,4789 52,103969 53,4909 54,4969 55,104149 57,1039694019687845983054132464844800000000000000000000000000000000003469 58,5209 59,78382080003589 [/CODE] 35:{0}:x:49 [CODE] 1,7560109 2,2269 3,126229 4,2389 5,126349 6,16456474460160000000000000409 8,453600529 9,2689 10,2749 11,5878656000000709 12,76187381760000000000769 13,97977600000829 15,3049 16,3109 17,3169 18,3229 19,127189 20,127249 22,3469 23,3529 24,27855313884812403095371776000000000000000000000000000001489 25,127549 26,3709 27,3769 29,3889 30,127849 31,7561909 32,7561969 33,4129 34,(trivial factor of 59) 36,453602209 37,1632960002269 38,7562329 39,128389 40,4549 41,128509 43,4729 44,4789 45,128749 46,4909 47,4969 48,352719360000002929 50,129049 51,5209 52,129169 53,129229 54,129289 55,5449 57,5569 58,129529 59,5689 [/CODE] 42:{0}:x:49 [CODE] 1,9072109 2,2689 3,2749 4,32659200289 5,151549 6,151609 8,3049 9,3109 10,3169 11,3229 12,151969 13,152029 15,3469 16,3529 17,117573120001069 18,5614347706314368308492315310161920000000000000000000000000000000000001129 19,3709 20,3769 22,3889 23,152629 24,9073489 25,16850366041100048016273224228350264013299690448515611349014072815152914037687200443877077133470937907200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,4129 27,(trivial factor of 59) 29,152989 30,5485491486720000000001849 31,32659201909 32,544321969 33,4549 34,9074089 36,4729 37,4789 38,153529 39,4909 40,4969 41,117573120002509 43,20686430901833768712610953618533187671699305261361528832000000000000000000000000000000000000000000000000000000000000000002629 44,5209 45,153949 46,544322809 47,544322869 48,5449 50,5569 51,387215843042271258003015621585831529981559389785592810438298620409934449687415027238102244809934838667668586191368584982641890866197562403890778944989683725107200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003109 52,5689 53,5749 54,32659203289 55,5869 57,154669 58,1959552003529 59,154789 [/CODE] 49:{0}:x:49 [CODE] 1,3049 2,3109 3,3169 4,3229 5,3379480093071544116029035238523182050306352376786936240751182230865067894959768623789993478122963368981956666059054838206787747840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349 6,176809 8,3469 9,3529 10,10584649 11,177109 12,3709 13,3769 15,3889 16,177409 17,137168640001069 18,140390781979454511600673751040000000000000000000000000000001129 19,4129 20,(trivial factor of 59) 22,635041369 23,38102401429 24,177889 25,177949 26,4549 27,178069 29,4729 30,4789 31,10585909 32,4909 33,4969 34,178489 36,178609 37,5209 38,635042329 39,1382343854653440000000000002389 40,10586449 41,5449 43,5569 44,179089 45,5689 46,5749 47,179269 48,5869 50,137168640003049 51,29628426240000003109 52,106662334464000000003169 53,635043229 54,6229 55,179749 57,10587469 58,6469 59,6529 [/CODE][/QUOTE] Other possible such simple families: (since x:0:(69-x):49 has trivial factor of 59) x:{0}:(69-x):(69-x):49 [CODE] 14,3225349 21,176989 28,47394646445260800000000000150109 35,250489 42,32659298869 49,402236156424545502745212987027034204727486491193140838400000000000000000000000000000000000000000000000000000000000000073249 [/CODE] |
There are still these families:
* x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) * 46:46:{42}:49 * x:46:{42}:49 (x is divisible by 7) * 46:x:{42}:49 (x is divisible by 7) |
3 Attachment(s)
Update the sieve files for the "minimal primes problem" in base 36
|
[QUOTE=sweety439;564441]There are still these families:
* x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) * 46:46:{42}:49 * x:46:{42}:49 (x is divisible by 7) * 46:x:{42}:49 (x is divisible by 7)[/QUOTE] Simple family 46:46:{42}:49 has prime 168409 For the family x:46:{42}:49 (x is divisible by 7): [CODE] 14,3192169 21,47407121744705084745769 28,17401277249084745769 35,100156988745769 42,9240169 49,179209 [/CODE] For the family 46:x:{42}:49 (x is divisible by 7): [CODE] 14,599337769 21,166909 28,167329 35,10064569 42,(trivial factor of 53, same as 46:{42}:49 49,7865395004745769 [/CODE] |
[QUOTE=sweety439;564440]Other possible such simple families: (since x:0:(69-x):49 has trivial factor of 59)
x:{0}:(69-x):(69-x):49 [CODE] 14,3225349 21,176989 28,47394646445260800000000000150109 35,250489 42,32659298869 49,402236156424545502745212987027034204727486491193140838400000000000000000000000000000000000000000000000000000000000000073249 [/CODE][/QUOTE] The corresponding number should contain x:{0}:(69-x):(69-x):49, thus the exponent n of x*60^n+(69-x)*3600+(69-x)*60+49 must be >=3 thus the numbers should be: [CODE] 14,3225349 21,34864577508285536715621843874728423450542982178891555528152012131493522929364204109502946145026849325046917080053260518080831302972748957603118253490089245371424805240428862727380360124313862377115462690866801581738795217778114560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000175729 28,47394646445260800000000000150109 35,7684489 42,32659298869 49,402236156424545502745212987027034204727486491193140838400000000000000000000000000000000000000000000000000000000000000073249 [/CODE] |
[QUOTE=sweety439;564430]x:{14}:49
[CODE] 1,109 2,8089 3,229 4,15289 5,349 6,409 8,1779289 9,33289 10,132675289 11,709 12,769 13,829 15,20055578263423416146440677966101694915289 16,1009 17,1069 18,1129 19,249315289 20,1249 22,1037502915289 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,6531289 31,6747289 32,116089 33,2029 34,2089 36,130489 37,2269 38,495555289 39,2389 40,144889 41,103182884390444449640082580739273165590541602171762902235022709553359196234458364272060077833460415740804281702234241751684562368942580406237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915289 43,155689 44,2689 45,2749 46,465965333694915289 47,10203289 48,37509315289 50,3049 51,3109 52,3169 53,3229 54,702915289 55,11931289 57,3469 58,3529 59,213289 [/CODE] x:{21}:49 [CODE] 1,109 2,508909 3,229 4,56452909 5,349 6,409 8,30109 9,94286240542372909 10,37309 11,709 12,769 13,829 15,42986782372909 16,1009 17,1069 18,1129 19,69709 20,1249 22,4025817357839737065999049857469767607759032979106716427631968228584180718644067796610169491525423728813559322033898305084745762711864406779661016949152542372909 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,66078257013152542372909 31,112909 32,6988909 33,2029 34,2089 36,101773342372909 37,2269 38,497092909 39,2389 40,31380772909 41,535972909 43,156109 44,2689 45,2749 46,166909 47,170509 48,10444909 50,3049 51,3109 52,3169 53,3229 54,195709 55,717412909 57,3469 58,3529 59,12820909 [/CODE] x:{28}:49 [CODE] 1,109 2,8929 3,229 4,57990529 5,349 6,409 8,30529 9,34129 10,2262529 11,709 12,769 13,829 15,3342529 16,1009 17,1069 18,1129 19,4206529 20,1249 22,80929 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,394950529 31,113329 32,116929 33,2029 34,2089 36,472710529 37,2269 38,8310529 39,2389 40,8742529 41,4212156168942609355932203389830529 43,9390529 44,2689 45,2749 46,167329 47,1041447436333870756881355932203389830529 48,13781461655186262022942372881355932203389830529 50,3049 51,3109 52,3169 53,3229 54,331946136337921041355932203389830529 55,199729 57,3469 58,3529 59,214129 [/CODE] x:{35}:49 [CODE] 1,109 2,9349 3,229 4,771484637288149 5,349 6,409 8,30949 9,34549 10,38149 11,709 12,769 13,829 15,56149 16,1009 17,1069 18,1129 19,70549 20,1249 22,81349 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,6608149 31,113749 32,25344488149 33,2029 34,2089 36,131749 37,2269 38,23335844534237288149 39,2389 40,526088149 41,149749 43,33898088149 44,2689 45,2749 46,10064149 47,185120325561205169675135454687984165581220276782105983361193164168670838265752820260904448577571678108045870876576322502449572547744055070802594515527926156691477915758601548938853501303980626387044701079429027352148995318212751455646534728345092572787387347410490834215458172781888350653835992601161966471095319325576754886993561238893441826145675686736477047030489460058140333352185261905102459753937730302319495540099005728377120233422356413887591971257315898743429492383226804915072909057608763313496683212160248130680520291356694770983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288149 48,10496149 50,3049 51,3109 52,3169 53,3229 54,196549 55,12008149 57,3469 58,3529 59,12872149 [/CODE] x:{42}:49 [CODE] 1,109 2,9769 3,229 4,170945053505084745769 5,349 6,409 8,1881769 9,2097769 10,38569 11,709 12,769 13,829 15,56569 16,1009 17,1069 18,1129 19,70969 20,1249 22,81769 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,110569 31,6849769 32,7065769 33,2029 34,2089 36,132169 37,2269 38,139369 39,2389 40,31657545769 41,150169 43,33990345769 44,2689 45,2749 46,(trivial factor of 53) 47,10305769 48,1767252099905084745769 50,3049 51,3109 52,3169 53,3229 54,11817769 55,200569 57,3469 58,3529 59,12897769 [/CODE] {14}:x:49 [CODE] 1,3074509 2,31719478566747843550804041300349830508474576271186440677966101694914569 3,11070914629 4,11070914689 5,184514749 6,11070914809 8,50929 9,50989 10,3075049 11,51109 12,51169 13,51229 15,51349 16,3075409 17,8608743701694915469 18,516524622101694915529 19,11070915589 20,184515649 22,51769 23,51829 24,664254915889 25,51949 26,52009 27,52069 29,52189 30,52249 31,39855294916309 32,52369 33,3076429 34,52489 36,52609 37,664254916669 38,30991477326101694916729 39,3076789 40,11070916849 41,664254916909 43,184517029 44,53089 45,53149 46,53690481146394350663097813232332427347712798372881355932203389830508474576271186440677966101694917209 47,53269 48,664254917329 50,86756301967596040677966101694917449 51,11070917509 52,53569 53,53629 54,3077689 55,3147493094329085095522234576271186440677966101694917749 57,30991477326101694917869 58,184517929 59,39855294917989 [/CODE] {21}:x:49 [CODE] 1,75709 2,10041238653656949152542371769 3,4611829 4,276771889 5,16606371949 6,4612009 8,76129 9,616626663337578078627630562878800705084745762711864406779661016949152542372189 10,76249 11,645936492161087054050399981653446997706215532953486454915466633416169490161247457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372309 12,76369 13,996382372429 15,4612549 16,4612609 17,4612669 18,276772729 19,4612789 20,4612849 22,4612969 23,77029 24,276773089 25,16606373149 26,4613209 27,77269 29,276773389 30,4613449 31,77509 32,77569 33,4613629 34,77689 36,16606373809 37,276773869 38,77929 39,16606373989 40,78049 41,4614109 43,78229 44,59782942374289 45,4614349 46,6071553036900241310806779661016949152542374409 47,276774469 48,61187265753757414256952240162711864406779661016949152542374529 50,78649 51,774786933152542374709 52,996382374769 53,4614829 54,78889 55,52209837304507200077039765027430616624964848906126524755206192493399488812789496523127382219681677062442951316647311186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542374949 57,4615069 58,996382375129 59,276775189 [/CODE] {28}:x:49 [CODE] 1,81583021005009885675936320216949152542372881355932203389828909 2,85460868854823834608016221386772928746969615932515587546395079909292691525423728813559322033898305084745762711864406779661016949152542372881355932203389828969 3,22141829029 4,101089 5,101149 6,101209 8,6149329 9,6149389 10,101449 11,286958123389829509 12,10410756236111524881355932203389829569 13,369029629 15,101749 16,79710589829809 17,101869 18,101929 19,1328509829989 20,6150049 22,13388318204875932203389830169 23,102229 24,6150289 25,79710589830349 26,102409 27,1328509830469 29,6150589 30,6150649 31,29828045081330194812832118462406904082062665762711864406779661016949152542372881355932203389830709 32,102769 33,102829 34,286958123389830889 36,41126317117261134141412615065804034201951786857394034478949601447152874939439946846072866224064662010793220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389831009 37,103069 38,6151129 39,1328509831189 40,369031249 41,369031309 43,286958123389831429 44,1328509831489 45,103549 46,1033049244203389831609 47,103669 48,6151729 50,1328509831849 51,286958123389831909 52,103969 53,37478722450001489572881355932203389832029 54,104089 55,104149 57,6152269 58,1328509832329 59,6152389 [/CODE] {35}:x:49 [CODE] 1,81475573154959817379255597463815552667176545024717505666213665960638224320021112459781503325113716943629003292963749951159320447728997238620350548107901686289125472897903137985503182233361374542375162900682265823294796999737443473337843435573308433456959404707058320810457306411455933006101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237286109 2,1660637286169 3,126229 4,17128518426043835517434182302188908474576271186440677966101694915254237286289 5,126349 6,36429318221401447864840677966101694915254237286409 8,77478693315254237286529 9,7686589 10,7686649 11,27677286709 12,99638237286769 13,461286829 15,126949 16,1660637287009 17,461287069 18,1660637287129 19,127189 20,127249 22,607155303690024131080677966101694915254237287369 23,131145545597045212313426440677966101694915254237287429 24,36429318221401447864840677966101694915254237287489 25,127549 26,127609 27,127669 29,27677287789 30,127849 31,358697654237287909 32,461287969 33,21521859254237288029 34,99638237288089 36,7688209 37,1660637288269 38,1660637288329 39,128389 40,128449 41,128509 43,128629 44,461288689 45,128749 46,7688809 47,358697654237288869 48,7688929 50,129049 51,7689109 52,129169 53,129229 54,129289 55,21521859254237289349 57,129469 58,129529 59,129589 [/CODE] {42}:x:49 [CODE] 1,9223309 2,293439587902707111288657562346647653394212382542384936694711075914089081641468094666645578615570550709924319586579296659981779875032195661701462202096102293874185480241052744144833349365537276799785969205297493379562404092952314178395004440542139367761173708310416661107311234222306240316209631923293822273260390016005704126405409242393725556515881932954120677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084743369 3,151429 4,260268905902788122033898305084743489 5,151549 6,151609 8,151729 9,95158435700243530652412123901049491525423728813559322033898305084743789 10,151849 11,151909 12,151969 13,152029 15,334707955121898305084744149 16,33212744209 17,9224269 18,25826231105084744329 19,152389 20,33212744449 22,1992764744569 23,152629 24,9224689 25,33212744749 26,152809 27,553544869 29,152989 30,1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049 31,3452320032561365140374087785000799083572067796610169491525423728813559322033898305084745109 32,33212745169 33,1992764745229 34,9225289 36,153409 37,153469 38,153529 39,153589 40,153649 41,9225709 43,202385101230008043693559322033898305084745829 44,153889 45,153949 46,430437185084746009 47,9226069 48,1992764746129 50,553546249 51,33212746309 52,154369 53,1204948638438833898305084746429 54,95158435700243530652412123901049491525423728813559322033898305084746489 55,9226549 57,154669 58,33212746729 59,154789 [/CODE][/QUOTE] x:{56}:49 [CODE] 1,109 2,232903489510723542345762711864406779661009 3,229 4,64141009 5,349 6,409 8,6958861009 9,35809 10,39409 11,709 12,769 13,829 15,2678845179661009 16,1009 17,1069 18,1129 19,71809 20,1249 22,82609 23,1429 24,1489 25,1549 26,1609 27,1669 29,1789 30,111409 31,24843661009 32,25621261009 33,2029 34,2089 36,6206038779661009 37,2269 38,2505535405903813370724804403593323757937224648962120170600421596767457627118644067796610169491525423728813559322033898305084745762711864406779661009 39,2389 40,147409 41,151009 43,158209 44,2689 45,2749 46,169009 47,1352698292914071864406779661009 48,3005939172097353755247425084745762711864406779661009 50,3049 51,3109 52,3169 53,3229 54,712141009 55,265108420301689108028745762711864406779661009 57,3469 58,3529 59,36248960086779661009 [/CODE] {56}:x:49 [CODE] 1,201709 2,201769 3,201829 4,201889 5,(unknown)-------------------------------------------- 6,12118009 8,202129 9,9653019592485633973013465317098305084745762711864406779661016989 10,2620643796649 11,202309 12,566400676881355969 13,5709735191002302915254237288135629 15,202549 16,58945898613104908796787281723327929495504791864406779661016949152542372881355932203389830508474609 17,567254718915254269 18,202729 19,12165589 20,12169249 22,568108760949152569 23,12180229 24,2046421359945762711889 25,2047036270210169491549 26,203209 27,7093761567185965006603828947492636122315332510451660944126786645957854271249499254305367632322963084970458439570233744677754294570009344165309519479915483916060404477260724290563504102526168108734063115285915290869621533585153192590815828858408818278898724882400084482921884840835304669997229790232023533140916445257577809774331239487911472434979629754359717806594801121257152551530794852095579580308470842508145255117645170320709345089612366101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050869 29,122969754676067796610189 30,203449 31,1188311170943907960149478894148859142785502131611931784122256194336006619664718924985760162565866809427612081912641567081289109267479099879367407659887603855337724488646981970874823867396949529636127713094173227389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169509 32,203569 33,2638831728829 34,12220489 36,203809 37,203869 38,443886502387850847457627129 39,203989 40,12530020127900548285031371932203389830508474576271186449 41,12533767143489035291097469830508474576271186440677966109 43,44112325429 44,(unknown)-------------------------------------------- 45,9533955905084749 46,12264409 47,445081887941857627118644069 48,2159152280280878917332772737414035022878368881841808054237288135593220338983050847457627118644067796610169491525423729 50,2652274983049 51,2063023937084745762709 52,44230942369 53,9781221482636363793361365248867796610169491525423728813559322029 54,44257301689 55,12297349 57,205069 58,205129 59,12311989 [/CODE] |
[QUOTE=sweety439;564438]14:{0}:x:49
[CODE] 1,24253982091278071092686802139899494400000000000000000000000000000000000000000109 2,1009 3,1069 4,1129 5,3024349 6,1249 8,50929 9,1429 10,1489 11,1549 12,1609 13,1669 15,1789 16,5361922889755312850308759177923802244504459923744908273254400000000000000000000000000000000000000000000000000000000000000000000001009 17,3025069 18,3025129 19,2029 20,2089 22,51769 23,2269 24,10886401489 25,2389 26,52009 27,52069 29,52189 30,2689 31,2749 32,52369 33,181442029 34,52489 36,3049 37,3109 38,3169 39,3229 40,39191040002449 41,3026509 43,3469 44,3529 45,53149 46,181442809 47,3709 48,3769 50,3889 51,181443109 52,53569 53,53629 54,4129 55,(trivial factor of 59) 57,3027469 58,307117308965289984000000000000000003529 59,10886403589 [/CODE] 21:{0}:x:49 [CODE] 1,75709 2,1429 3,1489 4,1549 5,1609 6,1669 8,1789 9,211631616000000589 10,76249 11,272160709 12,2029 13,2089 15,16329600949 16,2269 17,4537069 18,2389 19,4537189 20,272161249 22,12697896960000001369 23,2689 24,2749 25,2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,3527193600001609 27,77269 29,3049 30,3109 31,3169 32,3229 33,272162029 34,77689 36,3469 37,3529 38,77929 39,16329602389 40,3709 41,3769 43,3889 44,272162689 45,4538749 46,278553138848124030953717760000000000000000000000000002809 47,4129 48,(trivial factor of 59) 50,78649 51,12697896960000003109 52,4539169 53,979776003229 54,4549 55,58786560003349 57,4729 58,4789 59,592433080565760000000000003589 [/CODE] 28:{0}:x:49 [CODE] 1,1789 2,789910774087680000000000000169 3,21772800229 4,101089 5,2029 6,2089 8,21772800529 9,2269 10,101449 11,2389 12,6048769 13,6048829 15,101749 16,2689 17,2749 18,101929 19,47394646445260800000000000001189 20,362881249 22,3049 23,3109 24,3169 25,3229 26,102409 27,78382080001669 29,3469 30,3529 31,80223303988259720914670714880000000000000000000000000000001909 32,102769 33,3709 34,3769 36,3889 37,103069 38,282175488000002329 39,6050389 40,4129 41,(trivial factor of 59) 43,1306368002629 44,2316350688374295151333383964863082569625926687057800374045900800000000000000000000000000000000000000000000000000000000000000000000000002689 45,103549 46,362882809 47,4549 48,21772802929 50,4729 51,4789 52,103969 53,4909 54,4969 55,104149 57,1039694019687845983054132464844800000000000000000000000000000000003469 58,5209 59,78382080003589 [/CODE] 35:{0}:x:49 [CODE] 1,7560109 2,2269 3,126229 4,2389 5,126349 6,16456474460160000000000000409 8,453600529 9,2689 10,2749 11,5878656000000709 12,76187381760000000000769 13,97977600000829 15,3049 16,3109 17,3169 18,3229 19,127189 20,127249 22,3469 23,3529 24,27855313884812403095371776000000000000000000000000000001489 25,127549 26,3709 27,3769 29,3889 30,127849 31,7561909 32,7561969 33,4129 34,(trivial factor of 59) 36,453602209 37,1632960002269 38,7562329 39,128389 40,4549 41,128509 43,4729 44,4789 45,128749 46,4909 47,4969 48,352719360000002929 50,129049 51,5209 52,129169 53,129229 54,129289 55,5449 57,5569 58,129529 59,5689 [/CODE] 42:{0}:x:49 [CODE] 1,9072109 2,2689 3,2749 4,32659200289 5,151549 6,151609 8,3049 9,3109 10,3169 11,3229 12,151969 13,152029 15,3469 16,3529 17,117573120001069 18,5614347706314368308492315310161920000000000000000000000000000000000001129 19,3709 20,3769 22,3889 23,152629 24,9073489 25,16850366041100048016273224228350264013299690448515611349014072815152914037687200443877077133470937907200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,4129 27,(trivial factor of 59) 29,152989 30,5485491486720000000001849 31,32659201909 32,544321969 33,4549 34,9074089 36,4729 37,4789 38,153529 39,4909 40,4969 41,117573120002509 43,20686430901833768712610953618533187671699305261361528832000000000000000000000000000000000000000000000000000000000000000002629 44,5209 45,153949 46,544322809 47,544322869 48,5449 50,5569 51,387215843042271258003015621585831529981559389785592810438298620409934449687415027238102244809934838667668586191368584982641890866197562403890778944989683725107200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003109 52,5689 53,5749 54,32659203289 55,5869 57,154669 58,1959552003529 59,154789 [/CODE] 49:{0}:x:49 [CODE] 1,3049 2,3109 3,3169 4,3229 5,3379480093071544116029035238523182050306352376786936240751182230865067894959768623789993478122963368981956666059054838206787747840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349 6,176809 8,3469 9,3529 10,10584649 11,177109 12,3709 13,3769 15,3889 16,177409 17,137168640001069 18,140390781979454511600673751040000000000000000000000000000001129 19,4129 20,(trivial factor of 59) 22,635041369 23,38102401429 24,177889 25,177949 26,4549 27,178069 29,4729 30,4789 31,10585909 32,4909 33,4969 34,178489 36,178609 37,5209 38,635042329 39,1382343854653440000000000002389 40,10586449 41,5449 43,5569 44,179089 45,5689 46,5749 47,179269 48,5869 50,137168640003049 51,29628426240000003109 52,106662334464000000003169 53,635043229 54,6229 55,179749 57,10587469 58,6469 59,6529 [/CODE][/QUOTE] the exponent n must be >=2 thus the numbers should be: 14:{0}:x:49 [CODE] 1,24253982091278071092686802139899494400000000000000000000000000000000000000000109 2,85310363601469440000000000000000169 3,181440229 4,181440289 5,3024349 6,3024409 8,50929 9,50989 10,3024649 11,51109 12,51169 13,51229 15,51349 16,5361922889755312850308759177923802244504459923744908273254400000000000000000000000000000000000000000000000000000000000000000000001009 17,3025069 18,3025129 19,10886401189 20,39191040001249 22,51769 23,51829 24,10886401489 25,51949 26,52009 27,52069 29,52189 30,52249 31,3025909 32,52369 33,181442029 34,52489 36,52609 37,181442269 38,3026329 39,3026389 40,39191040002449 41,3026509 43,30474952704000000002629 44,53089 45,53149 46,181442809 47,53269 48,3026929 50,8465264640000003049 51,181443109 52,53569 53,53629 54,519847009843922991527066232422400000000000000000000000000000000003289 55,(trivial factor of 59) 57,3027469 58,307117308965289984000000000000000003529 59,10886403589 [/CODE] 21:{0}:x:49 [CODE] 1,75709 2,272160169 3,1658433468412565913600000000000000000000229 4,16329600289 5,82957725632529349183096088094078902389028226882976538255839640308244599894942626191769600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349 6,761873817600000000409 8,76129 9,211631616000000589 10,76249 11,272160709 12,76369 13,979776000829 15,16329600949 16,58786560001009 17,4537069 18,272161129 19,4537189 20,272161249 22,12697896960000001369 23,77029 24,16329601489 25,2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,3527193600001609 27,77269 29,1658433468412565913600000000000000000001789 30,2742745743360000000001849 31,77509 32,77569 33,272162029 34,77689 36,3527193600002209 37,12697896960000002269 38,77929 39,16329602389 40,78049 41,4538509 43,78229 44,272162689 45,4538749 46,278553138848124030953717760000000000000000000000000002809 47,2742745743360000000002869 48,(trivial factor of 59) 50,78649 51,12697896960000003109 52,4539169 53,979776003229 54,78889 55,58786560003349 57,16329603469 58,3527193600003529 59,592433080565760000000000003589 [/CODE] 28:{0}:x:49 [CODE] 1,6048109 2,789910774087680000000000000169 3,21772800229 4,101089 5,101149 6,101209 8,21772800529 9,13165179568128000000000000589 10,101449 11,6048709 12,6048769 13,6048829 15,101749 16,21772801009 17,101869 18,101929 19,47394646445260800000000000001189 20,362881249 22,362881369 23,102229 24,6049489 25,21772801549 26,102409 27,78382080001669 29,21772801789 30,1306368001849 31,80223303988259720914670714880000000000000000000000000000001909 32,102769 33,102829 34,6050089 36,362882209 37,103069 38,282175488000002329 39,6050389 40,78382080002449 41,(trivial factor of 59) 43,1306368002629 44,2316350688374295151333383964863082569625926687057800374045900800000000000000000000000000000000000000000000000000000000000000000000000002689 45,103549 46,362882809 47,103669 48,21772802929 50,1306368003049 51,614234617930579968000000000000000003109 52,103969 53,78382080003229 54,104089 55,104149 57,1039694019687845983054132464844800000000000000000000000000000000003469 58,362883529 59,78382080003589 [/CODE] 35:{0}:x:49 [CODE] 1,7560109 2,8295772563252934918309608809407890238902822688297653825583964030824459989494262619176960000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000169 3,126229 4,453600289 5,126349 6,16456474460160000000000000409 8,453600529 9,16456474460160000000000000589 10,1632960000649 11,5878656000000709 12,76187381760000000000769 13,97977600000829 15,126949 16,7561009 17,453601069 18,453601129 19,127189 20,127249 22,21163161600000001369 23,1058618139930670449693711081043149246240742123451547245555903649209833331097600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001429 24,27855313884812403095371776000000000000000000000000000001489 25,127549 26,127609 27,127669 29,7561789 30,127849 31,7561909 32,7561969 33,464255231413540051589529600000000000000000000000000002029 34,(trivial factor of 59) 36,453602209 37,1632960002269 38,7562329 39,128389 40,128449 41,128509 43,128629 44,5878656000002689 45,128749 46,7562809 47,7562869 48,352719360000002929 50,129049 51,1269789696000000003109 52,129169 53,129229 54,129289 55,1632960003349 57,129469 58,129529 59,129589 [/CODE] 42:{0}:x:49 [CODE] 1,9072109 2,25395793920000000169 3,151429 4,32659200289 5,151549 6,151609 8,151729 9,9072589 10,151849 11,151909 12,151969 13,152029 15,9072949 16,32659201009 17,117573120001069 18,5614347706314368308492315310161920000000000000000000000000000000000001129 19,152389 20,9073249 22,9073369 23,152629 24,9073489 25,16850366041100048016273224228350264013299690448515611349014072815152914037687200443877077133470937907200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 26,152809 27,(trivial factor of 59) 29,152989 30,5485491486720000000001849 31,32659201909 32,544321969 33,5485491486720000000002029 34,9074089 36,153409 37,153469 38,153529 39,153589 40,153649 41,117573120002509 43,20686430901833768712610953618533187671699305261361528832000000000000000000000000000000000000000000000000000000000000000002629 44,153889 45,153949 46,544322809 47,544322869 48,9074929 50,9075049 51,387215843042271258003015621585831529981559389785592810438298620409934449687415027238102244809934838667668586191368584982641890866197562403890778944989683725107200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003109 52,154369 53,14772772476559131803121204930915190318100750753041978023597954641499975561510557076017704836828037656902825724306702668559522982093578013910983164901324528981969784421602291230795234581969764786884845389890123210865111407518668131288178601114344226068215563391858075723302927140211618251554048985252761126263361099239362046694128659460701378499098943318291633703276353429221375899179448599101811449547970964749313875828661985567305524048394452992000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003229 54,32659203289 55,25395793920000003349 57,154669 58,1959552003529 59,154789 [/CODE] 49:{0}:x:49 [CODE] 1,176509 2,635040169 3,176629 4,23580260366520346895667724302680064000000000000000000000000000000000000000289 5,3379480093071544116029035238523182050306352376786936240751182230865067894959768623789993478122963368981956666059054838206787747840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349 6,176809 8,10584529 9,176989 10,10584649 11,177109 12,8230118400000769 13,8230118400000829 15,635040949 16,177409 17,137168640001069 18,140390781979454511600673751040000000000000000000000000000001129 19,177589 20,(trivial factor of 59) 22,635041369 23,38102401429 24,177889 25,177949 26,493807104000001609 27,178069 29,10585789 30,178249 31,10585909 32,10585969 33,383984404070400000000002029 34,178489 36,178609 37,10586269 38,635042329 39,1382343854653440000000000002389 40,10586449 41,178909 43,179029 44,179089 45,38102402749 46,179209 47,179269 48,38102402929 50,137168640003049 51,29628426240000003109 52,106662334464000000003169 53,635043229 54,179689 55,179749 57,10587469 58,635043529 59,179989 [/CODE] 56:{0}:x:49 [CODE] 1,201709 2,201769 3,201829 4,201889 5,9405849600000349 6,73708154151669596160000000000000000000409 8,202129 9,12096589 10,2031663513600000000649 11,202309 12,1579821548175360000000000000769 13,(trivial factor of 59) 15,202549 16,43545601009 17,12097069 18,202729 19,43545601189 20,15920961296760632770560000000000000000000001249 22,12097369 23,12097429 24,438839318937600000000001489 25,43545601549 26,203209 27,393769398563209972074510068169201449481098556918352173620112148964244949549133907206190467334984828895355316871076009117083636780520923662813598354443998931321515584635671891365271355834108131954442039194796452662249576087338013973280335640663780129596229993141561665027264712661995223080071542562945855136153140271526716432372295748613901305709330327823699068831518910889066496000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001669 29,725761789 30,203449 31,12097909 32,203569 33,725762029 34,156764160002089 36,203809 37,203869 38,725762329 39,203989 40,7485796941752491077989753746882560000000000000000000000000000000000002449 41,43545602509 43,725762629 44,156764160002689 45,7313988648960000000002749 46,12098809 47,43545602869 48,7485796941752491077989753746882560000000000000000000000000000000000002929 50,12099049 51,725763109 52,140121763265938537447937873720739226926078690331074652274769844785983881775031005160249279556824173287142353778948596279311875900939698176000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003169 53,12099229 54,73708154151669596160000000000000000003289 55,12099349 57,205069 58,205129 59,12099589 [/CODE] |
[QUOTE=sweety439;564441]There are still these families:
* x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) * 46:46:{42}:49 * x:46:{42}:49 (x is divisible by 7) * 46:x:{42}:49 (x is divisible by 7)[/QUOTE] For the families * x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) the formula is x*60^n+(69-x)*3600+y*60+49, and the exponent n must be >=3 [CODE] 14,14,30474952704000000198889 14,21,3223309 14,28,3223729 14,35,39191040200149 14,42,507915878400000200569 14,49,181640989 14,56,181641409 21,14,4709689 21,21,3527193600174109 21,28,16329774529 21,35,4710949 21,42,592433080565760000000000175369 21,49,4711789 21,56,366636387458016977907728122767084862675353600000000000000000000000000000000000000000000000176209 28,14,282175488000148489 28,21,6196909 28,28,282175488000149329 28,35,21772949749 28,42,6198169 28,49,1015831756800000150589 28,56,811595095103989852199754334506901252037658952536546715570377242634958349479237785790235037396561504839643390299093296141925888112898373496415048016409202885079874806616375810473697072981470141047462017636334871127550253811801712618213586079953084484967040245525358051328000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000151009 35,14,27216123289 35,21,27216123709 35,28,5878656000124129 35,35,7684549 35,42,5878656000124969 35,49,7685389 35,56,7685809 42,14,544418089 42,21,544418509 42,28,9170929 42,35,32659299349 42,42,9171769 42,49,7054387200100189 42,56,9172609 49,14,1382343854653440000000000072889 49,21,635113309 49,28,635113729 49,35,10658149 49,42,10658569 49,49,38102474989 49,56,10659409 56,14,12143689 56,21,43545648109 56,28,5687357573431296000000000000048529 56,35,9405849600048949 56,42,121899810816000000049369 56,49,15920961296760632770560000000000000000000049789 56,56,2612736050209 [/CODE] |
[QUOTE=sweety439;564727]For the families
* x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) the formula is x*60^n+(69-x)*3600+y*60+49, and the exponent n must be >=3 [CODE] 14,14,30474952704000000198889 14,21,3223309 14,28,3223729 14,35,39191040200149 14,42,507915878400000200569 14,49,181640989 14,56,181641409 21,14,4709689 21,21,3527193600174109 21,28,16329774529 21,35,4710949 21,42,592433080565760000000000175369 21,49,4711789 21,56,366636387458016977907728122767084862675353600000000000000000000000000000000000000000000000176209 28,14,282175488000148489 28,21,6196909 28,28,282175488000149329 28,35,21772949749 28,42,6198169 28,49,1015831756800000150589 28,56,811595095103989852199754334506901252037658952536546715570377242634958349479237785790235037396561504839643390299093296141925888112898373496415048016409202885079874806616375810473697072981470141047462017636334871127550253811801712618213586079953084484967040245525358051328000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000151009 35,14,27216123289 35,21,27216123709 35,28,5878656000124129 35,35,7684549 35,42,5878656000124969 35,49,7685389 35,56,7685809 42,14,544418089 42,21,544418509 42,28,9170929 42,35,32659299349 42,42,9171769 42,49,7054387200100189 42,56,9172609 49,14,1382343854653440000000000072889 49,21,635113309 49,28,635113729 49,35,10658149 49,42,10658569 49,49,38102474989 49,56,10659409 56,14,12143689 56,21,43545648109 56,28,5687357573431296000000000000048529 56,35,9405849600048949 56,42,121899810816000000049369 56,49,15920961296760632770560000000000000000000049789 56,56,2612736050209 [/CODE][/QUOTE] the family x:{0}:y:(69-x):49 [CODE] 14,14,1828497162240000000053749 14,21,3102949 14,28,87314335528601055933672487703638179840000000000000000000000000000000000000000104149 14,35,23697323222630400000000000129349 14,42,653184154549 14,49,10886579749 14,56,1421839393357824000000000000204949 21,14,4589329 21,21,471497411854445702041831433599646171136000000000000000000000000000000000000000000078529 21,28,272263729 21,35,761873817600000128929 21,42,272314129 21,49,4715329 21,56,979776204529 28,14,47394646445260800000000000052909 28,21,21772878109 28,28,1306368103309 28,35,1015831756800000128509 28,42,6201709 28,49,363058909 28,56,6252109 35,14,7612489 35,21,453677689 35,28,453702889 35,35,453728089 35,42,1034321545091688435630547680926659383584965263068076441600000000000000000000000000000000000000000000000000000000000000000153289 35,49,453778489 35,56,7763689 42,14,9124069 42,21,1535565253221389061599916669552741110897456632189955692192083681474829512272135129453510922525361221690477166314271723033120188802017840969135872571387935377488607242892909787410267763689425664432463903752046585965934041592221392289259925461699783631470003583650680234108357136242629037178594940076711634387530300209453418542748194570067935682926374815357451554271990640558506156240804286643522510255672570190067061858896266865292979492043550504659720216667943567422442954337288192000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000077269 42,28,32659302469 42,35,544447669 42,42,117573120152869 42,49,32659378069 42,56,32659403269 49,14,137168640051649 49,21,3869678092962653798400000000000000000076849 49,28,137168640102049 49,35,10711249 49,42,237634695574640633829083042534221670252544000000000000000000000000000000000000000000000152449 49,49,10761649 49,56,1382343854653440000000000202849 56,14,566535196960657629448499885738460574873117973650423757487464494926592599020095087950406759426690829879145778405826838856593058244296318326568243605199392115332255985874049174289486092523862589770124229194573292876887438699211884282069693077913600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051229 56,21,564350976000076429 56,28,2612736101629 56,35,(unknown) 56,42,438839318937600000000152029 56,49,12273229 56,56,43545802429 [/CODE] |
Now, the unsolved families are:
{56}:5:49 (not needed, since 5 is prime) {56}:44:49 (very low weight but eventually should yield a prime) 56:{0}:35:13:49 (not needed, since 13 is prime) |
[QUOTE=sweety439;564729]Now, the unsolved families are:
{56}:5:49 (not needed, since 5 is prime) {56}:44:49 (very low weight but eventually should yield a prime) 56:{0}:35:13:49 (not needed, since 13 is prime)[/QUOTE] No, 44:49 is prime (equals 2689), thus {56}:44:49 also not needed. |
[QUOTE=sweety439;564727]For the families
* x:{0}:(69-x):y:49 (both x and y are divisible by 7) * x:{0}:y:(69-x):49 (both x and y are divisible by 7) the formula is x*60^n+(69-x)*3600+y*60+49, and the exponent n must be >=3 [CODE] 14,14,30474952704000000198889 14,21,3223309 14,28,3223729 14,35,39191040200149 14,42,507915878400000200569 14,49,181640989 14,56,181641409 21,14,4709689 21,21,3527193600174109 21,28,16329774529 21,35,4710949 21,42,592433080565760000000000175369 21,49,4711789 21,56,366636387458016977907728122767084862675353600000000000000000000000000000000000000000000000176209 28,14,282175488000148489 28,21,6196909 28,28,282175488000149329 28,35,21772949749 28,42,6198169 28,49,1015831756800000150589 28,56,811595095103989852199754334506901252037658952536546715570377242634958349479237785790235037396561504839643390299093296141925888112898373496415048016409202885079874806616375810473697072981470141047462017636334871127550253811801712618213586079953084484967040245525358051328000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000151009 35,14,27216123289 35,21,27216123709 35,28,5878656000124129 35,35,7684549 35,42,5878656000124969 35,49,7685389 35,56,7685809 42,14,544418089 42,21,544418509 42,28,9170929 42,35,32659299349 42,42,9171769 42,49,7054387200100189 42,56,9172609 49,14,1382343854653440000000000072889 49,21,635113309 49,28,635113729 49,35,10658149 49,42,10658569 49,49,38102474989 49,56,10659409 56,14,12143689 56,21,43545648109 56,28,5687357573431296000000000000048529 56,35,9405849600048949 56,42,121899810816000000049369 56,49,15920961296760632770560000000000000000000049789 56,56,2612736050209 [/CODE][/QUOTE] Note that x:{0}:(69-x):0:49 and x:{0}:0:(69-x):49 do not need to test, since they have trivial factor of 59 |
Thus, we have proved that {40}[SUB]1937[/SUB]:1 is the largest minimal prime in base 60, but.... we should check the number {40}:x:1 and x:{40}:1
{40}:x:1 [CODE] 0,8784001 1,144061 2,527184121 3,527184181 4,144241 5,8784301 6,113872271184361 7,527184421 8,144481 9,144541 10,31631184601 11,527184661 12,33913043196358664526990564042370209820226038068458566486664714249719322033898305084745762711864406779661016949152542372881355932203389830508474576271184721 13,527184781 14,31631184841 15,527184901 16,144961 17,145021 18,113872271185081 19,149880623539480242643915932203389830508474576271185141 20,527185201 21,8785261 22,318769481068474576271185321 23,145381 24,145441 25,145501 26,409940176271185561 27,1147570131846508474576271185621 28,145681 29,31631185741 30,8785801 31,145861 32,5312824684474576271185921 33,8785981 34,192747715457150517803389830508474576271186041 35,68854207910790508474576271186101 36,146161 37,146221 38,527186281 39,113872271186341 41,5312824684474576271186461 42,146521 43,146581 44,8786641 45,146701 46,527186761 47,1897871186821 48,1897871186881 49,146941 50,8787001 51,8787061 52,113872271187121 53,1897871187181 54,88547078074576271187241 55,8787301 56,1288837126602692024758924485248572936113417984425523353957090008979918809550003571313773092508713399484305998216665053288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271187361 57,5312824684474576271187421 58,147481 59,147541 [/CODE] x:{40}:1 [CODE] 1,61 2,9601 3,181 4,241 5,57220816271186401 6,24001 7,421 8,6906499132699249581031646155932203389830508474576271186401 9,541 10,601 11,661 12,5960989295980474576271186401 13,49201 14,1490693601268614508474576271186401 15,56401 16,216146401 17,1021 18,242066401 19,15301586401 20,1201 21,4682401 22,1321 23,1381 24,88801 25,92401 26,96001 27,1621 28,629107126839310717830508474576271186401 29,1741 30,1801 31,1861 32,5488643471186401 33,7274401 34,7490401 35,99875471186401 36,2161 37,2221 38,2281 39,2341 41,150001 42,2521 43,9434401 44,9650401 45,2131151186401 46,1733242935968721451168935113911972881355932203389830508474576271186401 47,10298401 48,38741143572484853118599390155932203389830508474576271186401 49,3122357147620671464157533543332665566590202619661016949152542372881355932203389830508474576271186401 50,3001 51,3061 52,3121 53,3181 54,11810401 55,3301 56,3361 57,747506401 58,45627986401 59,3541 [/CODE] |
[QUOTE=sweety439;564728]the family x:{0}:y:(69-x):49
[CODE] 14,14,1828497162240000000053749 14,21,3102949 14,28,87314335528601055933672487703638179840000000000000000000000000000000000000000104149 14,35,23697323222630400000000000129349 14,42,653184154549 14,49,10886579749 14,56,1421839393357824000000000000204949 21,14,4589329 21,21,471497411854445702041831433599646171136000000000000000000000000000000000000000000078529 21,28,272263729 21,35,761873817600000128929 21,42,272314129 21,49,4715329 21,56,979776204529 28,14,47394646445260800000000000052909 28,21,21772878109 28,28,1306368103309 28,35,1015831756800000128509 28,42,6201709 28,49,363058909 28,56,6252109 35,14,7612489 35,21,453677689 35,28,453702889 35,35,453728089 35,42,1034321545091688435630547680926659383584965263068076441600000000000000000000000000000000000000000000000000000000000000000153289 35,49,453778489 35,56,7763689 42,14,9124069 42,21,1535565253221389061599916669552741110897456632189955692192083681474829512272135129453510922525361221690477166314271723033120188802017840969135872571387935377488607242892909787410267763689425664432463903752046585965934041592221392289259925461699783631470003583650680234108357136242629037178594940076711634387530300209453418542748194570067935682926374815357451554271990640558506156240804286643522510255672570190067061858896266865292979492043550504659720216667943567422442954337288192000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000077269 42,28,32659302469 42,35,544447669 42,42,117573120152869 42,49,32659378069 42,56,32659403269 49,14,137168640051649 49,21,3869678092962653798400000000000000000076849 49,28,137168640102049 49,35,10711249 49,42,237634695574640633829083042534221670252544000000000000000000000000000000000000000000000152449 49,49,10761649 49,56,1382343854653440000000000202849 56,14,566535196960657629448499885738460574873117973650423757487464494926592599020095087950406759426690829879145778405826838856593058244296318326568243605199392115332255985874049174289486092523862589770124229194573292876887438699211884282069693077913600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051229 56,21,564350976000076429 56,28,2612736101629 56,35,(unknown) 56,42,438839318937600000000152029 56,49,12273229 56,56,43545802429 [/CODE][/QUOTE] 1535565253221389061599916669552741110897456632189955692192083681474829512272135129453510922525361221690477166314271723033120188802017840969135872571387935377488607242892909787410267763689425664432463903752046585965934041592221392289259925461699783631470003583650680234108357136242629037178594940076711634387530300209453418542748194570067935682926374815357451554271990640558506156240804286643522510255672570190067061858896266865292979492043550504659720216667943567422442954337288192000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000077269 has 1096 digits but not minimal prime in base 60, since 21:27:49 (=77269) and 27:49 (=1669) are primes |
Base 60 minimal primes with >=500 decimal digits:
[CODE] {26}_(896):1 = 44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161 {40}_(1937):1 = 77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 47:{35}_(686):49 = 185120325561205169675135454687984165581220276782105983361193164168670838265752820260904448577571678108045870876576322502449572547744055070802594515527926156691477915758601548938853501303980626387044701079429027352148995318212751455646534728345092572787387347410490834215458172781888350653835992601161966471095319325576754886993561238893441826145675686736477047030489460058140333352185261905102459753937730302319495540099005728377120233422356413887591971257315898743429492383226804915072909057608763313496683212160248130680520291356694770983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288149 {35}_(367):1:49 = 81475573154959817379255597463815552667176545024717505666213665960638224320021112459781503325113716943629003292963749951159320447728997238620350548107901686289125472897903137985503182233361374542375162900682265823294796999737443473337843435573308433456959404707058320810457306411455933006101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237286109 {42}_(455):2:49 = 293439587902707111288657562346647653394212382542384936694711075914089081641468094666645578615570550709924319586579296659981779875032195661701462202096102293874185480241052744144833349365537276799785969205297493379562404092952314178395004440542139367761173708310416661107311234222306240316209631923293822273260390016005704126405409242393725556515881932954120677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084743369 {42}_(786):30:49 = 1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049 {49}_(437):15:49 = 3370877628514757222772400494683673158596409477797247719358711849483581786873706143158543146664730075705489224200224093772041630182187652701699645946277442899767500755812009374102776457086216749247209626019900398116606709317609816539188162397141065837083072359107434141070736000398492419524277805139751989637641458276851772548616402672230694356610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932201349 21:{0}_(313):25:49 = 2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 21:{0}_(289):48:48:49 = 34864577508285536715621843874728423450542982178891555528152012131493522929364204109502946145026849325046917080053260518080831302972748957603118253490089245371424805240428862727380360124313862377115462690866801581738795217778114560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000175729 56:{27}_(562):49 = 7093761567185965006603828947492636122315332510451660944126786645957854271249499254305367632322963084970458439570233744677754294570009344165309519479915483916060404477260724290563504102526168108734063115285915290869621533585153192590815828858408818278898724882400084482921884840835304669997229790232023533140916445257577809774331239487911472434979629754359717806594801121257152551530794852095579580308470842508145255117645170320709345089612366101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050869 42:{0}_(568):53:49 = 14772772476559131803121204930915190318100750753041978023597954641499975561510557076017704836828037656902825724306702668559522982093578013910983164901324528981969784421602291230795234581969764786884845389890123210865111407518668131288178601114344226068215563391858075723302927140211618251554048985252761126263361099239362046694128659460701378499098943318291633703276353429221375899179448599101811449547970964749313875828661985567305524048394452992000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003229 56:{0}:(481):27:49 = 393769398563209972074510068169201449481098556918352173620112148964244949549133907206190467334984828895355316871076009117083636780520923662813598354443998931321515584635671891365271355834108131954442039194796452662249576087338013973280335640663780129596229993141561665027264712661995223080071542562945855136153140271526716432372295748613901305709330327823699068831518910889066496000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001669 28:{0}_(342):41:56:49 = 811595095103989852199754334506901252037658952536546715570377242634958349479237785790235037396561504839643390299093296141925888112898373496415048016409202885079874806616375810473697072981470141047462017636334871127550253811801712618213586079953084484967040245525358051328000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000151009 42:{0}_(612):21:27:49 = 1535565253221389061599916669552741110897456632189955692192083681474829512272135129453510922525361221690477166314271723033120188802017840969135872571387935377488607242892909787410267763689425664432463903752046585965934041592221392289259925461699783631470003583650680234108357136242629037178594940076711634387530300209453418542748194570067935682926374815357451554271990640558506156240804286643522510255672570190067061858896266865292979492043550504659720216667943567422442954337288192000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000077269 56:{0}_(308):14:13:49 = 566535196960657629448499885738460574873117973650423757487464494926592599020095087950406759426690829879145778405826838856593058244296318326568243605199392115332255985874049174289486092523862589770124229194573292876887438699211884282069693077913600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051229 [/CODE] |
[QUOTE=sweety439;564747]Base 60 minimal primes with >=500 decimal digits:
[CODE] {26}_(896):1 = 44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161 {40}_(1937):1 = 77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 47:{35}_(686):49 = 185120325561205169675135454687984165581220276782105983361193164168670838265752820260904448577571678108045870876576322502449572547744055070802594515527926156691477915758601548938853501303980626387044701079429027352148995318212751455646534728345092572787387347410490834215458172781888350653835992601161966471095319325576754886993561238893441826145675686736477047030489460058140333352185261905102459753937730302319495540099005728377120233422356413887591971257315898743429492383226804915072909057608763313496683212160248130680520291356694770983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288149 {35}_(367):1:49 = 81475573154959817379255597463815552667176545024717505666213665960638224320021112459781503325113716943629003292963749951159320447728997238620350548107901686289125472897903137985503182233361374542375162900682265823294796999737443473337843435573308433456959404707058320810457306411455933006101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237286109 {42}_(455):2:49 = 293439587902707111288657562346647653394212382542384936694711075914089081641468094666645578615570550709924319586579296659981779875032195661701462202096102293874185480241052744144833349365537276799785969205297493379562404092952314178395004440542139367761173708310416661107311234222306240316209631923293822273260390016005704126405409242393725556515881932954120677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084743369 {42}_(786):30:49 = 1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049 {49}_(437):15:49 = 3370877628514757222772400494683673158596409477797247719358711849483581786873706143158543146664730075705489224200224093772041630182187652701699645946277442899767500755812009374102776457086216749247209626019900398116606709317609816539188162397141065837083072359107434141070736000398492419524277805139751989637641458276851772548616402672230694356610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932201349 21:{0}_(313):25:49 = 2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549 21:{0}_(289):48:48:49 = 34864577508285536715621843874728423450542982178891555528152012131493522929364204109502946145026849325046917080053260518080831302972748957603118253490089245371424805240428862727380360124313862377115462690866801581738795217778114560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000175729 56:{27}_(562):49 = 7093761567185965006603828947492636122315332510451660944126786645957854271249499254305367632322963084970458439570233744677754294570009344165309519479915483916060404477260724290563504102526168108734063115285915290869621533585153192590815828858408818278898724882400084482921884840835304669997229790232023533140916445257577809774331239487911472434979629754359717806594801121257152551530794852095579580308470842508145255117645170320709345089612366101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050869 42:{0}_(568):53:49 = 14772772476559131803121204930915190318100750753041978023597954641499975561510557076017704836828037656902825724306702668559522982093578013910983164901324528981969784421602291230795234581969764786884845389890123210865111407518668131288178601114344226068215563391858075723302927140211618251554048985252761126263361099239362046694128659460701378499098943318291633703276353429221375899179448599101811449547970964749313875828661985567305524048394452992000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003229 56:{0}:(481):27:49 = 393769398563209972074510068169201449481098556918352173620112148964244949549133907206190467334984828895355316871076009117083636780520923662813598354443998931321515584635671891365271355834108131954442039194796452662249576087338013973280335640663780129596229993141561665027264712661995223080071542562945855136153140271526716432372295748613901305709330327823699068831518910889066496000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001669 28:{0}_(342):41:56:49 = 811595095103989852199754334506901252037658952536546715570377242634958349479237785790235037396561504839643390299093296141925888112898373496415048016409202885079874806616375810473697072981470141047462017636334871127550253811801712618213586079953084484967040245525358051328000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000151009 42:{0}_(612):21:27:49 = 1535565253221389061599916669552741110897456632189955692192083681474829512272135129453510922525361221690477166314271723033120188802017840969135872571387935377488607242892909787410267763689425664432463903752046585965934041592221392289259925461699783631470003583650680234108357136242629037178594940076711634387530300209453418542748194570067935682926374815357451554271990640558506156240804286643522510255672570190067061858896266865292979492043550504659720216667943567422442954337288192000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000077269 56:{0}_(308):14:13:49 = 566535196960657629448499885738460574873117973650423757487464494926592599020095087950406759426690829879145778405826838856593058244296318326568243605199392115332255985874049174289486092523862589770124229194573292876887438699211884282069693077913600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051229 [/CODE][/QUOTE] The 1st number {26}_(896):1 [B]is minimal prime[/B] The 2nd number {40}_(1937):1 [B]is minimal prime[/B] The 3rd number 47:{35}_(686):49 is not minimal prime, since 47 is prime The 4th number {35}_(367):1:49 is not minimal prime, since 35:35:35:1 (=7688101) and 1:49 (=109) are primes The 5th number {42}_(455):2:49 is not minimal prime, since 2 is prime The 6th number {42}_(786):30:49 [B]is minimal prime[/B] The 7th number {49}_(437):15:49 [B]is minimal prime[/B] The 8th number 21:{0}_(313):25:49 is not minimal prime, since 25:49 (=1549) is prime The 9th number 21:{0}_(289):48:48:49 [B]is minimal prime[/B] The 10th number 56:{27}_(562):49 is not minimal prime, since 27:49 (=1669) is prime The 11th number 42:{0}_(568):53:49 is not minimal prime, since 53 and 53:49 (=3229) are primes The 12th number 56:{0}:(481):27:49 is not minimal prime, since 27:49 (=1669) is prime The 13th number 28:{0}_(342):41:56:49 is not minimal prime, since 41 is prime The 14th number 42:{0}_(612):21:27:49 is not minimal prime, since 27:49 (=1669) is prime The 15th number 56:{0}_(308):14:13:49 is not minimal prime, since 13 and 13:49 (=829) are primes |
Thus, the top 5 minimal primes in base 60 are:
[CODE] {40}_(1937):1 = 77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 {26}_(896):1 = 44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161 {42}_(786):30:49 = 1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049 {49}_(437):15:49 = 3370877628514757222772400494683673158596409477797247719358711849483581786873706143158543146664730075705489224200224093772041630182187652701699645946277442899767500755812009374102776457086216749247209626019900398116606709317609816539188162397141065837083072359107434141070736000398492419524277805139751989637641458276851772548616402672230694356610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932201349 21:{0}_(289):48:48:49 = 34864577508285536715621843874728423450542982178891555528152012131493522929364204109502946145026849325046917080053260518080831302972748957603118253490089245371424805240428862727380360124313862377115462690866801581738795217778114560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000175729 [/CODE] They are the only "base 60 minimal primes" > 10^500 |
2 Attachment(s)
reserve the only two unsolved families in base 36 (O{L}Z and {P}SZ), started with n=49K (double checked 49K-50K)
|
The "minimal prime problem" is solved only in bases 2~16, 18, 20, 22~24, 30, 42, 60
[CODE] b, length of largest minimal prime base b, number of minimal primes base b 2, 2, 2 3, 3, 3 4, 2, 3 5, 5, 8 6, 5, 7 7, 5, 9 8, 9, 15 9, 4, 12 10, 8, 26 11, 45, 152 12, 8, 17 13, 32021, 228 14, 86, 240 15, 107, 100 16, 3545, 483 18, 33, 50 20, 449, 651 22, 764, 1242 23, 800874, 6021 24, 100, 306 30, 1024, 220 42, 487, 4551 60, 1938, ? (should check all minimal primes) [/CODE] Base 13 and 23 data based in the case that one allows probable primes in place of proven primes. |
[QUOTE=sweety439;564952]reserve the only two unsolved families in base 36 (O{L}Z and {P}SZ), started with n=49K (double checked 49K-50K)[/QUOTE]
O{L}Z at 67946 L's {P}SZ at 65187 P's both no (probable) prime found |
where A=10, B=11, C=12, ...
and "SUB" tag means repeating digits, e.g. 123[SUB]4[/SUB]567 = 123333567 |
[QUOTE=sweety439;564439][URL="https://raw.githubusercontent.com/xayahrainie4793/primes/master/kernel60.txt"]minimal primes in base 60 up to 2^32[/URL][/QUOTE]
The link moves to [URL="https://raw.githubusercontent.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes/master/kernel60.txt"]https://raw.githubusercontent.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes/master/kernel60.txt[/URL] |
The GitHub link for minimal primes and left truncatable primes and right truncatable primes is now [URL="https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes"]https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes[/URL]
|
[QUOTE=sweety439;565043]O{L}Z at 67946 L's
{P}SZ at 65187 P's both no (probable) prime found[/QUOTE] O{L}Z at 68945 Ls {P}SZ at 66001 Ps both no (probable) prime found |
[URL="https://scholar.colorado.edu/downloads/hh63sw661"]the old PDF page for bases 2 to 10[/URL]
|
[URL="https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf"]the newest PDF page, include the (probable) prime (51*21^479149-1243)/4[/URL]
|
[QUOTE=sweety439;564315]In [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski problem[/URL] base b, the prime for a k-value <b is "minimal prime base b" if and only if k is not prime.
In [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel problem[/URL] base b, the prime for a k-value <b is "minimal prime base b" if and only if neither k-1 nor b-1 is prime. However, if we exclude the single-digit primes from the set (i.e. the minimal string of the set of prime numbers >= b in base b, see problem [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), then the prime for Sierpinski/Riesel problems base b for a k-value <b is always "minimal prime base b", this is why the "minimal prime problem" for the prime numbers >= b in base b is more interesting, since single-digit primes are trivial, like that in Sierpinski/Riesel problems base b, n=0 is trivial, since the corresponding number is just k+1 or k-1, and thus CRUS requires n>=1, and of course the CRUS Sierpinski/Riesel problems (requiring n>=1) is much harder than the same problem which n=0 is allowed, similarly, finding the minimal set of the strings for primes in base b with at least two digits in base b is much harder than finding the minimal set of the strings for primes (including the single-digit primes in base b) in base b, e.g. * In base 7, the largest minimal prime is 11111, but if single-digit primes are excluded, then a much-larger prime 33333333333333331 is minimal prime. * In base 8, the largest minimal prime is 444444441, but if single-digit primes are excluded, then a much-larger prime 7777777777771 is minimal prime. * In base 10, the largest minimal prime is 66600049, but if single-digit primes are excluded, then a much-larger prime 555555555551 is minimal prime. * In base 14, the largest minimal prime is 40[SUB]83[/SUB]49, but if single-digit primes are excluded, then a much-larger prime 4D[SUB]19698[/SUB] is minimal prime. * In base 17, there are only 2 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 74[SUB]4904[/SUB] is minimal prime. * In base 21, there are only 3 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 5D0[SUB]19848[/SUB]1 is minimal prime. * In base 30, the largest minimal prime is C0[SUB]1022[/SUB]1, but if single-digit primes are excluded, then a much-larger prime OT[SUB]34205[/SUB] is minimal prime. * In base 32, there are 78 unsolved families when searched to length 10000, but if single-digit primes are excluded, then the unsolved family S{V} is searched up to length 2000001 by CRUS with no prime found. * In base 33, there are 33 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 130[SUB]23614[/SUB]1 is minimal prime. * In base 35, there are only 15 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 1B0[SUB]56061[/SUB]1 is minimal prime. * In base 37, if single-digit primes are excluded, then the unsolved family 2K{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family {I}J is searched up to length 524287 with no (probable) prime found. * In base 38, if single-digit primes are excluded, then there are four large known minimal primes 20[SUB]2728[/SUB]1, V0[SUB]1527[/SUB]1, Lb[SUB]1579[/SUB], and ab[SUB]136211[/SUB]. * In base 42, the largest minimal prime is R[SUB]486[/SUB]1, but if single-digit primes are excluded, then a much-larger prime 2f[SUB]2523[/SUB] is minimal prime. * In base 43, if single-digit primes are excluded, then the unsolved family 3b{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family 2{7} is searched up to length 50001 with no (probable) prime found. * In base 48, if single-digit primes are excluded, then there is a large known minimal prime T0[SUB]133041[/SUB]1. * In base 60, if single-digit primes are excluded, then the unsolved family Z{x} is searched up to length 100001 by CRUS with no prime found.[/QUOTE] If even the prime 10 (i.e. the prime equal to the base (b)) is excluded, this is the "minimal prime problem" for the prime numbers > b in base b and contain more primes (for prime base b, if b is not prime, then the "minimal prime problem" for the prime numbers (>=b and >b) are completely the same) * In base 19, a large prime F10[SUB]18523[/SUB]1 is minimal prime. * In base 29, a large prime 10[SUB]8095[/SUB]A is minimal prime. * In base 37, a large prime 1F0[SUB]1627[/SUB]1 is minimal prime. * In base 47, a large prime 10[SUB]112[/SUB]2 is minimal prime. * In base 53, a large prime 10[SUB]13401[/SUB]4 is minimal prime. * In base 53, 19{0}1 is unsolved family searched to length 305002. * In base 89, a large prime 10[SUB]254[/SUB]2 is minimal prime. * In base 107, a large prime 1:0[SUB]1399[/SUB]:(106) is minimal prime. * In base 113, a large prime 10[SUB]10645[/SUB]4 is minimal prime. * In base 113, a large prime 1:0[SUB]20087[/SUB]:(112) is minimal prime. * In base 139, 1{0}4 is unsolved family searched to length 25000. * In base 167, 1{0}2 is unsolved family searched to length 100001. |
There are three problems:
* Find the set of all minimal primes base b when single-digit prime substrings are allowed (see [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), for 2<=b<=256 * Find the set of all left-truncatable primes base b when the single-digit suffix (i.e. the rightmost digit) need not to be prime, for 2<=b<=256 * Find the set of all right-truncatable primes base b when the single-digit prefix (i.e. the leftmost digit) need not to be prime (see [URL="https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/17229#17229"]https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/17229#17229[/URL] and [URL="https://hlma.math.cuhk.edu.hk/wp-content/uploads/2018/06/a90bcf7cf0e95d023687faea1b2408fa.pdf"]https://hlma.math.cuhk.edu.hk/wp-content/uploads/2018/06/a90bcf7cf0e95d023687faea1b2408fa.pdf[/URL]), for 2<=b<=256 Compare with the original problems .... * Find the set of all minimal primes base b, for 2<=b<=256 * Find the set of all left-truncatable primes base b, for 2<=b<=256 * Find the set of all right-truncatable primes base b, for 2<=b<=256 |
[QUOTE=sweety439;566583]There are three problems:
* Find the set of all minimal primes base b when single-digit prime substrings are allowed (see [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), for 2<=b<=256 * Find the set of all left-truncatable primes base b when the single-digit suffix (i.e. the rightmost digit) need not to be prime, for 2<=b<=256 * Find the set of all right-truncatable primes base b when the single-digit prefix (i.e. the leftmost digit) need not to be prime (see [URL="https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/17229#17229"]https://codegolf.meta.stackexchange.com/questions/2140/sandbox-for-proposed-challenges/17229#17229[/URL] and [URL="https://hlma.math.cuhk.edu.hk/wp-content/uploads/2018/06/a90bcf7cf0e95d023687faea1b2408fa.pdf"]https://hlma.math.cuhk.edu.hk/wp-content/uploads/2018/06/a90bcf7cf0e95d023687faea1b2408fa.pdf[/URL]), for 2<=b<=256 Compare with the original problems .... * Find the set of all minimal primes base b, for 2<=b<=256 * Find the set of all left-truncatable primes base b, for 2<=b<=256 * Find the set of all right-truncatable primes base b, for 2<=b<=256[/QUOTE] (single-digit primes are not in these sets) The 1st set for b=10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, ...} The 2nd set for b=10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 131, 137, 167, 173, 179, 197, 211, 223, 229, 241, 271, 283, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 419, 431, 443, 461, 467, 479, 523, 541, 547, 571, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 683, 719, 743, 761, 773, 797, 811, 823, 829, 853, 859, 883, 911, 919, 929, 937, 941, 947, 953, 967, 971, 983, 997, 1223, 1229, 1283, 1367, 1373, 1523, 1571, 1613, 1619, 1811, 1823, 1997, 2113, 2131, 2137, 2179, 2311, 2347, 2383, 2389, 2467, 2617, 2647, 2659, 2683, 2719, 2797, 2953, 2971, 3137, 3167, 3229, 3271, 3313, 3331, 3347, 3359, 3373, 3389, 3461, 3467, 3541, 3547, 3571, 3613, 3617, 3631, 3643, 3659, 3673, 3719, 3761, 3797, 3823, 3853, 3911, 3919, 3929, 3947, 3967, 4211, 4229, 4241, 4271, 4283, 4337, 4373, 4397, 4523, 4547, 4643, 4673, 4919, 4937, 4967, 5113, 5167, 5179, 5197, 5347, 5419, 5431, 5443, 5479, 5641, 5647, 5653, 5659, 5683, 5743, 5953, 6113, 6131, 6173, 6197, 6211, 6229, 6271, 6311, 6317, 6337, 6353, 6359, 6367, 6373, 6379, 6389, 6397, 6547, 6571, 6619, 6653, 6659, 6661, 6673, 6719, 6761, 6823, 6829, 6883, 6911, 6947, 6967, 6971, 6983, 6997, 7211, 7229, 7283, 7331, 7523, 7541, 7547, 7643, 7673, 7823, 7829, 7853, 7883, 7919, 7937, 8167, 8179, 8311, 8317, 8353, 8389, 8419, 8431, 8443, 8461, 8467, 8641, 8647, 8719, 8761, 8929, 8941, 8971, 9137, 9173, 9241, 9283, 9311, 9337, 9397, 9419, 9431, 9461, 9467, 9479, 9547, 9613, 9619, 9631, 9643, 9661, 9719, 9743, 9811, 9829, 9859, 9883, 9929, 9941, 9967, 12113, 12347, 12647, 12659, 12953, 13229, 13313, 13331, 13613, 13967, 15443, 15641, 15647, 15683, 16229, 16547, 16619, 16661, 16673, 16823, 16829, 16883, 18311, 18353, 18443, 18461, 18719, 19661, 21283, 21523, 21613, 21997, 23167, 23719, 23761, 23911, 23929, 24229, 24337, 24373, 24547, 24919, 24967, 26113, 26317, 26947, 27211, 27283, 27541, 27673, 27823, 27883, 27919, 29137, 29173, 29311, 31223, 32467, 32647, 32719, 32797, 32971, 33331, 33347, 33359, 33461, 33547, 33613, 33617, 33797, 33911, 33967, 34211, 34283, 34337, 34673, 34919, 35419, 36131, 36229, 36353, 36373, 36389, 36571, 36653, 36761, 36947, 36997, 37547, 37643, 37853, 38167, 38317, 38431, 38461, 38971, 39241, 39397, 39419, 39461, 39619, 39631, 39719, 39829, 39883, 39929, 42131, 42179, 42467, 42683, 42719, 42797, 42953, 43271, 43313, 43331, 43541, 43613, 43853, 45179, 45197, 45641, 45659, 45953, 46229, 46271, 46337, 46619, 46829, 46997, 48179, 48311, 48353, 48647, 48761, 49547, 49613, 49811, 51229, 51283, 51613, 53359, 53617, 53719, 54547, 54673, 54919, 56113, 56131, 56197, 56311, 56359, 56659, 56911, 56983, 57283, 57331, 57829, 57853, 59419, 59467, 59743, 59929, 61223, 61283, 61613, 62131, 62137, 62311, 62347, 62383, 62467, 62617, 62659, 62683, 62971, 63313, 63331, 63347, 63389, 63467, 63541, 63617, 63659, 63719, 63761, 63823, 63853, 63929, 64271, 64283, 64373, 64919, 64937, 65167, 65179, 65419, 65479, 65647, 66173, 66271, 66337, 66359, 66373, 66571, 66653, 66883, 66947, 67211, 67523, 67547, 67829, 67853, 67883, 68311, 68389, 68443, 69337, 69431, 69467, 69661, 69829, 69859, 69929, 69941, 72383, 72467, 72617, 72647, 72719, 72797, 72953, 73331, 73547, 73571, 73613, 73643, 73673, 73823, 75167, 75347, 75431, 75479, 75641, 75653, 75659, 75683, 75743, 76367, 76379, 76673, 76829, 76883, 78167, 78179, 78311, 78317, 78467, 78929, 78941, 79241, 79283, 79337, 79397, 79613, 79631, 79811, 79829, 79967, 81223, 81283, 81373, 81619, 83137, 83389, 83617, 83719, 83761, 83911, 84211, 84229, 84523, 84673, 84919, 84967, 86113, 86131, 86197, 86311, 86353, 86389, 86719, 87211, 87523, 87541, 87547, 87643, 87853, 89137, 89431, 91229, 91283, 91367, 91373, 91571, 91811, 91823, 91997, 92179, 92311, 92347, 92383, 92467, 92647, 92683, 93229, 93719, 93761, 93911, 93967, 94229, 94397, 94547, 95419, 95443, 95479, 96211, 96337, 96353, 96661, 96823, 96911, 96997, 97283, 97523, 97547, 97673, 97829, 97883, 97919, 98179, 98317, 98389, 98419, 98443, 98467, 98641, 98929, 99137, 99173, 99241, 99397, 99431, 99643, 99661, 99719, 99829, 99859, 99929, ...} The 3rd set for b=10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 233, 239, 293, 311, 313, 317, 373, 379, 419, 431, 433, 439, 479, 593, 599, 613, 617, 619, 673, 677, 719, 733, 739, 797, 839, 971, 977, 1319, 1373, 1399, 1733, 1913, 1931, 1933, 1973, 1979, 1993, 1997, 1999, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 4337, 4339, 4391, 4397, 4793, 4799, 5939, 6131, 6133, 6173, 6197, 6199, 6733, 6737, 6779, 7193, 7331, 7333, 7393, 9719, 13997, 13999, 17333, 19139, 19319, 19333, 19739, 19793, 19937, 19973, 19979, 19991, 19993, 19997, 23333, 23339, 23399, 23993, 29399, 31193, 31379, 37337, 37339, 37397, 43391, 43397, 43399, 43913, 43973, 47933, 47939, 59393, 59399, 61331, 61333, 61339, 61979, 61991, 67339, 71933, 73331, 73939, 139991, 139999, 193337, 197933, 199373, 199379, 199739, 199799, 199931, 199933, 233993, 239933, 293999, 373379, 373393, 439133, 593933, 593993, 613337, 619793, 673391, 673397, 673399, 719333, 739391, 739393, 739397, 739399, ..., 1979339339} The 1st set for b=2 is {10, 11} The 2nd set for b=2 is {10, 11, 111} The 3rd set for b=2 is {10, 11, 101, 111, 1011, 10111, 101111} The 1st set for b=3 is {10, 12, 21, 111} The 2nd set for b=3 is {10, 12, 21, 212} The 3rd set for b=3 is {10, 12, 21, 102, 122, 212, 1222, 2122} The 1st set for b=4 is {11, 13, 23, 31, 221} The 2nd set for b=4 is {11, 13, 23, 31, 113, 131, 211, 223, 311, 323, 331, 1211, 1223, 2113, 2131, 2311, 3211, 3323, 21211, 21223, 32113, 33211, 33323, 121211, 233323, 321223, 333323, 2121211} The 3rd set for b=4 is {11, 13, 23, 31, 113, 131, 133, 233, 311, 1333, 2333, 13331, 133313} The 1st set for b=8 is {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, ...} The 2nd set for b=8 is {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 123, 145, 153, 213, 227, 235, 265, 323, 337, 345, 351, 357, 373, 415, 445, 475, 513, 521, 535, 557, 565, 573, 615, 621, 645, 657, 673, 715, 723, 737, 753, 775, 1145, 1153, 1357, 1475, 1737, 1775, 2213, 2235, 2521, 2535, 3123, 3145, 3235, 3323, 3337, 3373, 3513, 3521, 3615, 3673, 3715, 3723, 3753, 4123, 4351, 4357, 4445, 4753, 4775, 5213, 5227, 5265, 5345, 5521, 5535, 5557, 5573, 5615, 6235, 6265, 6345, 6373, 6475, 6615, 6715, 6723, 6775, 7153, 7357, 7415, 7445, 7673, 7723, 7775, 11737, 13323, 13615, 14775, 16265, 17357, 17415, 17673, 17723, 23123, 23145, 25213, 25227, 25557, 25573, 25615, 26723, 31153, 31775, 32213, 32235, 32521, 33123, 33615, 34123, 34445, 34753, 35345, 35573, 36265, 37357, 37415, 37723, 37775, 43235, 43323, 43337, 43513, 43521, 44357, 46615, 47445, 47673, 47775, 52235, 52521, 53723, 55345, 56475, 56723, 56775, 61145, 63521, 63715, 64357, 66715, 67415, 67673, 67723, 67775, 71357, 71475, 71737, 73337, 73513, 74123, 74445, 76345, 76615, 77153, 113323, 132213, 132235, 144357, 163715, 171357, 173337, 174445, 223123, 225573, 226723, 233615, 234753, 236265, 237357, 237415, 253723, 256775, 264357, 311737, 313323, 316265, 317673, 317723, 325557, 325615, 331153, 332235, 332521, 334753, 335345, 337723, 343235, 343337, 356475, 356723, 361145, 363521, 367415, 371357, 417415, 432235, 435573, 443513, 447673, 447775, 473337, 473513, 523123, 523145, 525557, 525615, 531153, 534123, 534445, 536265, 537357, 537415, 537723, 537775, 552521, 613615, 617415, 623123, 625227, 631775, 632521, 644357, 653723, 661145, 666715, 667415, 673513, 676345, 711737, 735573, 743337, 743521, 746615, 747775, 774123, 777153, ...} The 3rd set for b=8 is {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 131, 153, 155, 211, 213, 235, 277, 351, 357, 373, 513, 533, 535, 573, 577, 657, 737, 753, 1317, 1531, 1533, 1537, 1555, 2111, 2117, 2135, 2353, 2773, 3513, 3517, 3571, 3733, 5331, 5355, 5735, 5773, 6571, 7371, 7531, 7533, 15311, 15317, 15373, 15377, 15553, 21113, 21117, 21177, 21355, 23537, 27733, 27735, 35133, 35171, 35713, 37333, 53555, 73717, 153173, 153733, 153773, 211135, 211177, 277331, 277333, 351331, 351717, 535553, ...} The 1st set for b=12 is {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, ...} The 2nd set for b=12 is {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 111, 117, 11B, 125, 131, 13B, 145, 157, 167, 16B, 175, 181, 18B, 195, 1A7, 1B5, 1B7, 217, 21B, 225, 237, 24B, 251, 25B, 267, 285, 291, 295, 2AB, 315, 325, 327, 33B, 34B, 357, 35B, 375, 391, 3AB, 3B5, 3B7, 415, 41B, 427, 431, 435, 437, 457, 45B, 46B, 481, 485, 48B, 511, 517, 51B, 527, 531, 535, 545, 557, 575, 585, 587, 58B, 591, 5B5, 5B7, 611, 615, 617, 61B, 637, 63B, 661, 66B, 675, 687, 68B, 695, 6A7, 711, 71B, 727, 735, 737, 745, 751, 767, 76B, 775, 785, 791, 817, 825, 835, 851, 85B, 867, 881, 88B, 8A7, 8AB, 8B5, 8B7, 91B, 927, 95B, 987, 995, 9A7, 9AB, 9B5, A11, A17, A27, A35, A37, A3B, A45, A4B, A5B, A6B, A87, A91, A95, AA7, AAB, AB7, B11, B15, B1B, B25, B31, B37, B45, B61, B67, B6B, B91, B95, BB5, BB7, 1125, 1167, 118B, 11A7, 11B7, 121B, 125B, 1295, 133B, 1391, 13B5, 1431, 1437, 1457, 148B, 1517, 1585, 1587, 1591, 1615, 168B, 16A7, 1711, 1727, 1735, 1745, 1751, 176B, 1785, 1825, 18AB, 18B7, 1995, 1A11, 1A17, 1A35, 1A45, 1A6B, 1A87, 1AAB, 1AB7, 1B15, 1B67, 1BB5, 2111, 211B, 2131, 2181, 21A7, 21B7, 221B, 224B, 2267, 2291, 2325, 2327, 234B, 23AB, 23B7, 2415, 2435, 2457, 2481, 2485, 248B, 2535, 2545, 2557, 258B, 2615, 2617, 2637, 2675, 2687, 26A7, 2737, 2745, 276B, 2825, 2835, 285B, 2927, 29B5, 2A11, 2A35, 2A3B, 2A5B, 2A87, 2A95, 2AA7, 2B31, 2B61, 2B67, 2B95, 2BB7, 3117, 311B, 3145, 3167, 3175, 3195, 3225, 324B, 3285, 3291, 3327, 3357, 341B, 3427, 3435, 3457, 346B, 3481, 3485, 348B, 3517, 351B, 3587, 358B, 35B7, 3617, 3637, 3661, 366B, 3687, 3767, 3791, 3851, 38B5, 395B, 39A7, 3A11, 3A91, 3A95, 3AB7, 3B11, 3B1B, 3B61, 3BB7, 4111, 411B, 413B, 4145, 41B5, 4217, 4225, 4237, 4291, 42AB, 4357, 4375, 43B5, 4435, 445B, 4485, 4531, 4535, 4557, 4591, 4611, 4615, 463B, 4687, 468B, 4711, 4727, 4737, 4775, 4825, 4881, 488B, 48A7, 491B, 4987, 4A5B, 4A91, 4B15, 4B37, 4B95, 5117, 511B, 5131, 513B, 5145, 5167, 516B, 51B7, 521B, 5237, 525B, 5267, 5285, 5295, 52AB, 5327, 5375, 5391, 53AB, 53B5, 541B, 5435, 5457, 548B, 5527, 5531, 5545, 5585, 5587, 558B, 55B5, 5615, 5637, 563B, 566B, 56A7, 5711, 5727, 5735, 5785, 5817, 5835, 5867, 58AB, 58B5, 58B7, 5927, 595B, 5987, 59AB, 5A11, 5A17, 5A27, 5A45, 5A4B, 5A5B, 5A6B, 5A95, 5AAB, 5B1B, 5B25, 5B37, 5B67, 5B91, 5B95, 6117, 613B, 6175, 61A7, 61B7, 6327, 633B, 634B, 6357, 6375, 6391, 63B5, 63B7, 6437, 646B, 6527, 6575, 6591, 6617, 663B, 6675, 671B, 6751, 6825, 6881, 68B5, 6995, 69B5, 6A11, 6A17, 6A27, 6A4B, 6AAB, 6B15, 6B25, 7111, 711B, 7125, 7131, 716B, 7175, 718B, 71B5, 71B7, 7225, 7295, 7391, 73AB, 7415, 7427, 7435, 7457, 745B, 7511, 7531, 7585, 7587, 758B, 75B5, 7611, 7617, 761B, 7637, 763B, 7661, 766B, 7675, 7687, 771B, 7737, 7767, 776B, 7817, 7851, 7867, 795B, 79AB, 7A27, 7A35, 7A6B, 7A95, 7B11, 7B15, 7B25, 7B37, 7B67, 7B6B, 7B91, 8125, 816B, 8175, 8195, 81B7, 8251, 8291, 82AB, 833B, 8357, 835B, 83AB, 841B, 8427, 8431, 8511, 8517, 8575, 8591, 85B7, 8637, 866B, 8835, 8881, 888B, 88AB, 8995, 8A11, 8A37, 8A91, 8A95, 8AA7, 8B37, 8B45, 9131, 9145, 918B, 9195, 91A7, 9217, 9251, 9267, 92AB, 9315, 9375, 9485, 9557, 9575, 9591, 95B7, 9615, 9687, 9695, 9711, 9737, 9775, 9785, 988B, 98A7, 98B7, 991B, 9927, 99AB, 9A17, 9A35, 9AAB, 9B1B, 9B45, 9B61, A117, A13B, A145, A157, A1A7, A225, A24B, A285, A295, A315, A35B, A3B7, A457, A46B, A485, A511, A535, A575, A63B, A661, A675, A68B, A695, A711, A71B, A735, A745, A767, A76B, A791, A817, A825, A851, A85B, A88B, A987, A9B5, AA45, AA6B, AAB7, AB45, AB6B, AB91, ABB5, B125, B157, B167, B181, B18B, B1B5, B21B, B2AB, B315, B325, B327, B357, B3AB, B3B5, B481, B51B, B527, B5B7, B615, B617, B63B, B675, B68B, B711, B727, B785, B835, B85B, B8A7, B8B5, B91B, B987, B9A7, B9B5, BA27, BA4B, BA87, BBB7, ...} The 3rd set for b=12 is {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 111, 117, 11B, 157, 171, 175, 17B, 1B1, 1B5, 1B7, 251, 255, 25B, 271, 277, 27B, 315, 357, 35B, 375, 377, 3B5, 3B7, 455, 457, 45B, 4B1, 4BB, 511, 517, 51B, 575, 577, 5B1, 5B5, 5B7, 5BB, 611, 615, 617, 61B, 675, 6B1, 751, 817, 851, 855, 85B, 871, 8B5, 8B7, 91B, 955, 95B, A77, AB7, ABB, B71, 1115, 11B7, 1577, 157B, 1711, 1715, 1751, 1755, 1757, 17BB, 1B15, 1B17, 1B51, 1B7B, 2555, 2557, 2715, 2717, 2771, 27B1, 27B7, 3155, 315B, 35B1, 35B7, 35BB, 3755, 375B, 3771, 377B, 3B51, 3B55, 3B75, 3B7B, 4557, 455B, 4571, 4577, 457B, 4B15, 4BB1, 5117, 511B, 51B7, 575B, 5771, 5777, 577B, 5B17, 5B1B, 5B55, 5B75, 5BB1, 6115, 6117, 6171, 6175, 617B, 61B7, 6751, 6757, 675B, 6B15, 6B17, 7511, 8175, 8511, 8515, 8517, 85B7, 8717, 8B55, 8B71, 8B75, 9551, 9557, 95B7, A777, AB7B, ABB5, B711, 11151, 1115B, 11B71, 11B75, 15771, 157B1, 17115, 1711B, 17151, 17515, 17551, 17555, 17557, 1755B, 17571, 17575, 1757B, 17BB1, 1B155, 1B177, 1B17B, 1B517, 1B51B, 25551, 25577, 27151, 27155, 2715B, 27B17, 27B77, 31551, 315B5, 375B5, 375BB, 37715, 3B515, 3B557, 3B55B, 3B7B5, 4557B, 45775, 45777, 4B155, 4BB11, 4BB15, 511B7, 51B71, 575BB, 57711, 57717, 577B7, 577BB, 5B175, 5B1B7, 5B55B, 5B751, 5BB17, 61151, 6115B, 61755, 61757, 617B5, 617B7, 61B75, 61B77, 67517, 67575, 675B5, 6B151, 6B171, 75111, 75115, 85155, 85175, 85177, 8517B, 85B75, 85B7B, 8717B, 8B551, 8B555, 8B557, 8B711, 8B717, 8B757, 95511, 95517, 95575, A7775, A777B, AB7BB, ABB51, ABB5B, B7111, B7115, 111511, 11151B, 11B717, 11B71B, 15771B, 157B17, 171155, 171515, 175517, 175575, 17557B, 1755B7, 175715, 17575B, 1B1555, 1B1771, 1B1775, 1B17B1, 255515, 255775, 271555, 2715B1, 27B177, 27B17B, 27B771, 375B55, 375BB5, 377151, 3B5155, 3B5157, 3B515B, 3B5571, 3B557B, 3B55B7, 3B7B5B, 457771, 457775, 4B1551, 4BB155, 4BB157, 511B77, 51B717, 575BBB, 577117, 577175, 577B75, 5B55B1, 5B55BB, 5BB171, 611511, 61151B, 617557, 617575, 61757B, 617B75, 61B755, 675171, 675751, 675755, 675757, 675B51, 6B1711, 751115, 851557, 851751, 85175B, 8517B7, 8717B1, 8717BB, 8B555B, 8B5575, 8B7117, 8B7171, 8B717B, 8B7571, 955115, 955171, 955175, 955177, 955755, 95575B, A77755, AB7BB1, AB7BBB, ABB511, B71157, ...} |
Now my GitHub page [URL="https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes"]https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes[/URL] extended to include these numbers:
* [URL="https://primes.utm.edu/glossary/page.php?sort=MinimalPrime"]minimal primes[/URL] base b (file name: "kernel b"): Data is available for bases 2 to 81 and 84, 90, 96, 100, 108, 120, 126, 128, 144, 150 (for bases > 50 there is only such primes < 2^32 listed) (data for unsolved families (file name: "left b") is available for bases 2 to 50) * [URL="https://primes.utm.edu/glossary/page.php?sort=LeftTruncatablePrime"]left-truncatable primes[/URL] base b (file name: "ltp b"): Data is available for bases 2 to 45 (for bases 18, 20, 22, 24, 26, 28, 30, 32-36, 38-45 there is only such primes < 2^32 listed) * [URL="https://primes.utm.edu/glossary/page.php?sort=RightTruncatablePrime"]right-truncatable primes[/URL] base b (file name: "rtp b"): Data is available for bases 2 to 52 * two-sided primes (primes which are both left-truncatable and right-truncatable) base b (file name: "twoside b"): Data is available for bases 2 to 68 * minimal composites (composites in the sense of minimal primes) base b (file name: "mc b"): Data is available for bases 2 to 65 |
For minimal composites, I only searched to 4 digits, and I doubt that there will be some minimal composites with >=5 digits in some base
[B]Theorem: If base b has minimal composite with >=4 digits, then either b=2 or b is divisible by 6[/B] Proof: If b is odd, and there is a minimal composite with >=3 digit in base b, let string [I]xyz[/I] be any three-digit substring of this minimal composite number, since in odd base, [I]xy[/I] == [I]x[/I] + [I]y[/I] (mod 2), [I]xz[/I] == [I]x[/I] + [I]z[/I] (mod 2), [I]yz[/I] == [I]y[/I] + [I]z[/I] (mod 2), thus we have [I]xy[/I] + [I]xz[/I] + [I]yz[/I] == 2([I]x[/I] + [I]y[/I] + [I]z[/I]) == 0 (mod 2), and at least one of [I]xy[/I], [I]xz[/I], [I]yz[/I] must be even (0 is counted as even number), since all even numbers except 0 and 2 are composite, if all of [I]xy[/I], [I]xz[/I], [I]yz[/I] are noncomposite, then one of them must be 0 or 2, but if we let the [I]x[/I] be the leading digit, then [I]x[/I] cannot be 0, and neither [I]xy[/I] nor [I]xz[/I] can be 0 or 2 (since any odd base b is >=3, "base 1" does not exist, and 0, 2 are both single-digit number in base b, and hence written as "00", "02" for [I]xy[/I] and [I]xz[/I]), thus, [I]yz[/I] must be 0 or 2, and ([I]y[/I], [I]z[/I]) is either (0, 0) or (0, 2), and if [I]x[/I] is not 1, then [I]xy[/I] = [I]x[/I] * base (b), and is composite, thus [I]x[/I] must be 1, thus, [I]xyz[/I] is either 100 or 102, and the only such numbers (i.e. with leading digit 1 and all substrings containing the 1 are either 100 or 102) with n (>=4) digits are 1000...000 and 1000...0002 with n digits, however, both of them have 100 as substring, and 100 = base(b)^2 and hence composite, which is contradiction, [B]thus, for any odd base b, there are no minimal composites with >=3 digits, while 100 and 102 are the only two possible exceptions.[/B] If b is == 1 mod 3, let string [I]xyzw[/I] be any three-digit substring, then at least one of [I]xy[/I], [I]xz[/I], [I]yz[/I] is divisible by 3, unless ([I]x[/I], [I]y[/I], [I]z[/I]) == (0, 1, 1) or (0, 2, 2) (or their permutations) mod 3, and in this case, at least one of [I]0w[/I], [I]1w[/I] (or [I]2w[/I]), [I]11w[/I] (or [I]22w[/I]) is divisible by 3, thus [I]xyzw[/I] or any strings containing [I]xyzw[/I] cannot be minimal composites. (i.e. [B]for any base b == 1 mod 3, there are no minimal composites with >=4 digits[/B]) If b is == 2 mod 3, let string [I]xyzw[/I] be any three-digit substring, then at least two of [I]x[/I] mod 3, [I]y[/I] mod 3, [I]z[/I] mod 3, [I]w[/I] mod 3, are the same (by [URL="https://en.wikipedia.org/wiki/Pigeonhole_principle"]pigeonhole principle[/URL], 4 digits in 3 possible modulos), and the connection of these two digits are divisible by 3, thus [I]xyzw[/I] or any strings containing [I]xyzw[/I] cannot be minimal composites (the only exception is base 2, which the number divisible by 3 is exactly 11 (=3), and 1111 (=15) is minimal composite). (i.e. [B]for any base b == 2 mod 3 (except base 2), there are no minimal composites with >=4 digits[/B]) |
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