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Done for all bases <=50
Bases 43, 47, 49 are not listed as they have some unsolved families x{y}z with |x|>=7 or |z|>=7, thus my excel program cannot handle (will give error a or c value for (a*b^n+c)/d) |
(Probable) primes found for base 31:
[CODE] E8(U^21866)P = 443*31^21867-6 IE(L^29787) = (5727*31^29787-7)/10 L(F^21052)G = (43*31^21053+1)/2 MI(O^10747)L = (3504*31^10748-19)/5 PEO(0^22367)Q = 24483*31^22368+26 (R^22137)1R = (9*31^22139-8069)/10 [/CODE] Unsolved families searched to high depth with no (probable) prime found: [CODE] ILE(L^n) = (179637*31^n-7)/10 at n=30000 (now unneeded since IE(L^29787) is (probable) prime) L0(F^n)G = (1303*31^(n+1)+1)/2 at n=23000 (now unneeded since L(F^21052)G is (probable) prime) M(P^n) = (137*31^n-5)/6 at n=39000 P(F^n)G = (51*31^(n+1)+1)/2 at n=32000 (R^n)1 = (9*31^(n+1)-269)/10 at n=20000 (R^n)8 = (9*31^(n+1)-199)/10 at n=19000 (U^n)P8K = 31^(n+3)-5498 at n=27000 [/CODE] |
Newest status for the unsolved families in base 31:
(probable) primes found: [CODE] E8{U}P: prime at length 21869 (the prime is 443*31^21867-6) IE{L}: prime at length 29789 (the prime is (5727*31^29787-7)/10) L{F}G: prime at length 21054 (the prime is (43*31^21053+1)/2) MI{O}L: prime at length 10750 (the prime is (3504*31^10748-19)/5) PEO{0}Q: prime at length 22371 (the prime is 24483*31^22368+26) {L}9G: prime at length 10014 (the prime is (6727*31^10012-3777)/10) {R}1R: prime at length 22139 (the prime is (9*31^22139-8069)/10) [/CODE] unneeded families: [CODE] ILE{L} (tested to length 30000, but IE{L} has prime at length 29789) L0{F}G (tested to length 23000, but L{F}G has prime at length 21054) {L}9IG (tested to length 13000, but {L}9G has prime at length 10014) [/CODE] unsolved families: [CODE] M{P} (at length 39000) (the formula is (137*31^n-5)/6) P{F}G (at length 32000) (the formula is (1581*31^n+1)/2) SP{0}K (at length 28000) (the formula is 27683*31^n+20) {F}G (at length 4194303) (the formula is (31*31^n+1)/2) {F}KO (the formula is (961*31^n+327)/2) {F}RA (the formula is (961*31^n+733)/2) {L}CE (at length 21000) (the formula is (6727*31^n-2867)/10) {L}G (at length 30000) (the formula is (217*31^n-57)/10) {L}IS (at length 25000) (the formula is (6727*31^n-867)/10) {L}SO (at length 22000) (the formula is (6727*31^n+2193)/10) {P}I (at length 32000) (the formula is (155*31^n-47)/6) {R}1 (at length 27000) (the formula is (279*31^n-269)/10) {R}8 (at length 33000) (the formula is (8649*31^n-8069)/10) {U}P8K (at length 30000) (the formula is 29791*31^n-5498) [/CODE] |
If [URL="https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf"]1 is regarded as prime[/URL], then all bases <=20 are solved, in fact, all bases <=24 except the base 21 family G{0}FK is solved.
[CODE] b largest minimal prime 2 1 3 2 4 3 5 3 6 5 7 5 8 7 9 7 10 66600049 11 A999999999999999999999 12 B 13 940000000000000000000000000000000000C 14 40000000000000000000000000000000000000000000000000000000000000000000000000000000000049 15 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608 16 F88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888F 17 4(9^111333) 18 H 19 FG(6^110984) 20 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG99 21 ? (the smallest prime of the form G{0}FK if exists, otherwise C(F^479147)0K) 22 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAF 23 9(E^800873) 24 N [/CODE] However, for base 25, there are 6 unsolved families: [CODE] 6M{F}9 EF{O} F{0}KO F0{K}O LO{L}8 O{L}8 [/CODE] besides, for bases in [URL="https://oeis.org/A048597"]https://oeis.org/A048597[/URL], the largest minimal prime is the largest prime < base (i.e. the largest single-digit prime), since all digits coprime to the base are primes. |
For the minimal set of "noncomposites" (i.e. 0, 1, primes), all bases <=24 are completely solved:
[CODE] b largest minimal noncomposite 2 1 3 2 4 3 5 3 6 5 7 5 8 7 9 7 10 946669 11 A999999999999999999999 12 B 13 9866666666666666666666666666 14 99999999999999999999999999999999999989 15 E9666666666666668 16 F88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888F 17 4(9^111333) 18 H 19 FG(6^110984) 20 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG99 21 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA6FK 22 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAF 23 9(E^800873) 24 N [/CODE] and base 25 has 4 unsolved families: [CODE] 6M{F}9 EF{O} LO{L}8 O{L}8 [/CODE] |
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(5*36^81995+1)/7 (algebraic form)
(P^81995)SZ (base 36 form, with A=10, B=11, C=12, ...) is a [URL="https://primes.utm.edu/glossary/xpage/MinimalPrime.html"]minimal prime[/URL] in base 36, found by me this number is likely the second-largest minimal prime in base 36 see [URL="http://www.primenumbers.net/prptop/searchform.php?form=%285*36%5En%2B821%29%2F7&action=Search"]Top PRP page[/URL] |
Since some of large minimal primes in [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]https://github.com/curtisbright/mepn-data/tree/master/data[/URL] and [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] are only probable primes (i.e. not definitely primes), thus there is a possibility that |M(L[SUB]b[/SUB])| and max(x belong to M(L[SUB]b[/SUB]), |x|) are not the values in table in [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL]:
* For 2<=b<=12, 14<=b<=16, b=18, b=20, b=22, b=24, b=30, b=42, b=60, all minimal primes in base b are known and all these primes are definitely primes (i.e. not merely probable primes), thus we can definitely say that |M(L[SUB]b[/SUB])| and max(x belong to M(L[SUB]b[/SUB]), |x|) are the values in table * For b = 13, all minimal primes in base b are known and all these primes except 8(0^32017)111 are definitely primes, but 8(0^32017)111 is only probable prime, and there will be no unsolved family other than 8{0}111 if 8(0^32017)111 is composite (such family must contain 8{0}111 as sub-family), thus .... ** If 8(0^32017)111 is in fact prime (very likely, > 99.999999999999...%), then |M(L[SUB]b[/SUB])| = 228 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 32021 ** If 8(0^32017)111 is in fact composite but there is a larger prime of the form 8{0}111, then |M(L[SUB]b[/SUB])| = 228 and max(x belong to M(L[SUB]b[/SUB]), |x|) = length of the smallest such prime ** If 8(0^32017)111 is in fact composite and there is no prime of the form 8{0}111 (very unlikely), then |M(L[SUB]b[/SUB])| = 227 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 309 (the prime (9^308)1) (thus |M(L[SUB]b[/SUB])| is either 227 or 228 (and very likely 228) and max(x belong to M(L[SUB]b[/SUB]), |x|) is either 309 or >=32021 (and very likely 32021)) (based on probably prime tests, |M(L[SUB]b[/SUB])| = 228 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 32021) * For b = 17, there is an unsolved family F1{9} and all these primes except (6^4661)E9, 1(F^7092), 4(9^111333) are definitely primes, and (see [URL="https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9"]this page[/URL]) there will be no other unsolved families if these three numbers are composite), thus .... ** If (6^4661)E9, 1(F^7092), 4(9^111333) are in fact primes and there is a prime of the form F1{9} (very likely, > 99.999999999999...%), then |M(L[SUB]b[/SUB])| = 1280 and max(x belong to M(L[SUB]b[/SUB]), |x|) = length of the smallest prime of the form F1{9} ** If (6^4661)E9, 1(F^7092), 4(9^111333) are in fact primes but there is no prime of the form F1{9}, then |M(L[SUB]b[/SUB])| = 1279 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 111334 ** If only two of (6^4661)E9, 1(F^7092), 4(9^111333) are in fact primes and there is a prime of both the form F1{9} and the form corresponding to the composite, then |M(L[SUB]b[/SUB])| = 1280 and max(x belong to M(L[SUB]b[/SUB]), |x|) = the larger number in (length of the smallest prime of the form F1{9}, length of the smallest prime of the form corresponding to the composite) ** If only two of (6^4661)E9, 1(F^7092), 4(9^111333) are in fact primes but there is a prime of only one of the form F1{9} and the form corresponding to the composite, then |M(L[SUB]b[/SUB])| = 1279 and max(x belong to M(L[SUB]b[/SUB]), |x|) = the largest number in (lengths of the two primes in {(6^4661)E9, 1(F^7092), 4(9^111333)}, length of the smallest prime of the form F1{9} or the form corresponding to the composite) ** If none of (6^4661)E9, 1(F^7092), 4(9^111333) are in fact primes and there is no prime of either F1{9} or {6}E9, 1{F}, 4{9}, then |M(L[SUB]b[/SUB])| = 1276 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 2016 (the prime (A^2014)GF) (thus |M(L[SUB]b[/SUB])| is one of {1276, 1277, 1278, 1279, 1280} (and very likely 1280) and max(x belong to M(L[SUB]b[/SUB]), |x|) is either one of {2016, 4663, 7093, 111334} or >1000000 (the current searching limit of the length of F1{9})) (based on probably prime tests, |M(L[SUB]b[/SUB])| is either 1279 or 1280 and max(x belong to M(L[SUB]b[/SUB]), |x|) is either 111334 or >1000000 (the current searching limit of the length of F1{9})) * For b = 21, there is an unsolved family G{0}FK and all these primes except 4(0^47333)9G and C(F^479147)0K are definitely primes, and (see [URL="https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9"]this page[/URL]) there will be no other unsolved families if these three numbers are composite), thus .... ** If both 4(0^47333)9G and C(F^479147)0K are in fact primes and there is a prime of the form G{0}FK (very likely, > 99.999999999999...%), then |M(L[SUB]b[/SUB])| = 2601 and max(x belong to M(L[SUB]b[/SUB]), |x|) = length of the smallest prime of the form G{0}FK ** If neither 4(0^47333)9G nor C(F^479147)0K are in fact primes and there is no prime of either G{0}FK or 4{0}9G, C{F}0K, then |M(L[SUB]b[/SUB])| = 2598 and max(x belong to M(L[SUB]b[/SUB]), |x|) = 1634 (the prime (A^1631)6FK) (thus |M(L[SUB]b[/SUB])| is one of {2598, 2599, 2600, 2601} (and very likely 2601) and max(x belong to M(L[SUB]b[/SUB]), |x|) is either one of {1634, 47336, 479150} or > 475000 (the current searching limit of the length of G{0}FK)) * For b = 23, all minimal primes in base b are known and all these primes except the largest 9 primes (down to the prime FFF(K^4461)C, the next prime (K^3761)L is proven prime by N-1 test) (except the second-largest prime 8(0^119214)1, since this prime can be proven prime by N-1 test) are definitely primes, and see [URL="https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9"]https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9[/URL], and for the unsolved families, {G}69 will be another unsolved family if the prime for {G}9 is in fact composite (this is the only such example, since the prime for 9{E} is larger than the prime for 96{E} and K9A{E}, and 8{0}81 has no prime up to 119216 digits, but since 8(0^119214) is definitely prime (i.e. not merely probable prime), thus the prime for 8{0}81 is definitely not minimal prime), thus |M(L[SUB]b[/SUB])| is one of {6013, 6014, 6015, 6016, 6017, 6018, 6019, 6020, 6021, 6022} (based on probably prime tests, |M(L[SUB]b[/SUB])| = 6021), |M(L[SUB]b[/SUB])| = 6013 if none of the largest 9 primes except the second-largest prime 8(0^119214)1 are in fact primes and none of the corresponding families of these 8 composites contain a prime, |M(L[SUB]b[/SUB])| = 6022 if the prime for {G}9 is in fact composite and both families {G}9 and {G}69 contain a prime and the smallest prime of the form {G}69 is smaller than the smallest prime of the form {G}9, and max(x belong to M(L[SUB]b[/SUB]), |x|) is >= 119216 (since 8(0^119214)1 is definitely prime) and very likely 800874 (the prime 9(E^800873)), it is possible that one of the largest 9 primes is in fact composite and the corresponding family has the smallest prime > 119216 digits (also possible > 800874 digits), thus it is possible that in fact this family produces the largest minimal prime, and max(x belong to M(L[SUB]b[/SUB]), |x|) is > 800874 * For b = 26, all known minimal primes are definitely primes (i.e. not merely probable primes, the largest known minimal prime (M^8772)P has a primality certificate), but there are two unsolved families: {A}6F and {I}GL, and search limit of them is 461000 digits, thus |M(L[SUB]b[/SUB])| is one of {5662, 5663, 5664}, and max(x belong to M(L[SUB]b[/SUB]), |x|) is either 8773 or > 461000 |
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The largest known [URL="https://primes.utm.edu/glossary/xpage/MinimalPrime.html"]minimal (probable) prime[/URL] in [URL="https://archive.ph/nB56D"]base 36[/URL]: (P^81995)SZ (with A=10, B=11, C=12, ...)
this number is likely the second-largest minimal prime in base 36, [URL="https://raw.githubusercontent.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes/master/kernel36"]this[/URL] is the list of all known minimal (probable) primes in base 36, totally 6296 (probable) primes (except the last two, all are definitely primes (i.e. not merely probable primes), the last two are only probable primes), the only unsolved family is OLLL...LLLZ (i.e. if there is an additional minimal prime in base 36 not in the list, then it must be of the form OLLL...LLLZ, and of course there is at most one such prime), which was already searched to length 100000 by me. If we use decimal to write its digits, it will be (25^81995):28:35, and 81995 in base 36 is 1:27:9:21, thus the complete base 36 extension are (25^(1:27:9:21)):28:35 (note the numbers in the picture: 25:1:27:9:21:28:35, the "1" of one emerald does not shown, and the level is 36 (which means base 36), and thus it means (25^(1:27:9:21)):28:35 in base 36) And there is a 6*6 [URL="https://en.wikipedia.org/wiki/Latin_square"]Latin square[/URL] with 6 different colors of glasses (exactly 36 glasses) at the ceiling |
[QUOTE=sweety439;572226]Examples of section 4 in [URL="https://arxiv.org/pdf/1607.01548.pdf"]https://arxiv.org/pdf/1607.01548.pdf[/URL] (M(S intersection T) and M(S) union M(T)):
Let A := {primes written in decimal} B := {numbers > 10 written in decimal} C := {numbers == 1 mod 4 written in decimal} D := {numbers == 3 mod 4 written in decimal}[/QUOTE] Now I try to solve M(A intersection B intersection C) and M(A intersection B intersection D): For M(A intersection B intersection C), if the prime contains no 5, then the prime belongs to M(A intersection C), which is already in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/primes1mod4/minimal.10.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/primes1mod4/minimal.10.txt[/URL] (except the term 5), hence we assume that the prime contains at least one 5. ([B]bold[/B] for the primes containing at least one 5 in M(A intersection B intersection C)) Since the prime is == 1 mod 4, thus the final two digits of the prime are in {01, 09, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 81, 89, 93, 97} Case 1: The final two digits is 01 since 101, 41, 61, 701 are primes, thus we assume the prime is of the form {0,2,3,5,8,9}01 since the prime must contains at least one 5 and [B]5501[/B] is prime, thus we assume the prime contains exactly one 5 if the prime contains 8, then either [B]5801[/B] or [B]8501[/B] must be subsequence, thus we assume the prime contains no 8 thus the first digit of the prime is in {2,3,9} (note that the first digit cannot be 0) * If the first digit is 2, then we can write the prime as 2x5y01, and x,y only contain digits {0,2,3,9} both x and y cannot contain 9, or 29 will be a subsequence x cannot contain 2, or [B]22501[/B] will be a subsequence y cannot contain 2, or 521 will be a subsequence y cannot contain 3, or 53 will be a subsequence if x contain 3, then y contains no 0 (or 3001 will be a subsequence), y contains no 3 (or 233 will be a subsequence), thus y is empty, also x cannot contain two or more 3's (or 233 will be a subsequence), thus this prime belong to 2{0}3{0}501, and since 3001 is prime, we only need to check the families 2{0}3501, and the smallest prime of the form 2{0}3501 is [B]200003501[/B] thus the prime belong to 2{0}5{0}01 since [B]2005001[/B] is prime, we only need to check the families 25{0}01, 205{0}01, 2{0}501 ** the smallest prime of the form 25{0}01 is [B]2500000001[/B] ** the smallest prime of the form 205{0}01 is [B]205000001[/B] ** the smallest prime of the form 2{0}501 is [B]2000000000000501[/B] * If the first digit is 3, then we can write the prime as 3x5y01, and x,y only contain digits {0,2,3,9} both x and y cannot contain 0, or 3001 will be a subsequence both x and y cannot contain 3, or 3301 will be a subsequence y cannot contain 2, or 521 will be a subsequence x can only contain at most one 2, or 22501 will be a subsequence both x and y can contain at most one 9, and at most one of x and y can contain 9, or 9901 will be a subsequence thus the prime can only be {3501,32501,39501,35901,329501,392501,325901}, for these numbers only 325901 is prime, but 325901 is not minimal 4k+1 prime since it contains 29 as subsequence * If the first digit is 9, then we can write the prime as 9x5y01, and x,y only contain digits {0,2,3,9} both x and y cannot contain 0, or 9001 will be a subsequence both x and y cannot contain 9, or 9901 will be a subsequence both x and y can contain at most one 2, and at most one of x and y can contain 2, or 9221 will be a subsequence both x and y can contain at most one 3, and at most one of x and y can contain 3, or 3301 will be a subsequence thus the prime can only be {9501,92501,95201,93501,95301,923501,932501,925301,935201,952301,953201}, for these numbers only [B]923501[/B] and 935201 are primes, but 935201 is not minimal 4k+1 prime since it contains 521 as subsequence Case 2: The final two digits is 09 then the prime must contain [B]509[/B] as subsequence Case 3: The final two digits is 13 then the prime must contain 13 as subsequence Case 4: The final two digits is 17 then the prime must contain 17 as subsequence Case 5: The final two digits is 21 then the prime must contain [B]521[/B] as subsequence Case 6: The final two digits is 29 then the prime must contain 29 as subsequence Case 7: The final two digits is 33 then the prime must contain 53 as subsequence Case 8: The final two digits is 37 then the prime must contain 37 as subsequence Case 9: The final two digits is 41 then the prime must contain 41 as subsequence Case 10: The final two digits is 49 since 149, 29, 349, 449, 89 are primes, thus we assume the prime is of the form {0,5,6,7,9}49 if the prime contains 7, then either [B]5749[/B] or [B]7549[/B] must be subsequence, thus we assume the prime contains no 7 thus the first digit of the prime is in {5,6,9} (note that the first digit cannot be 0) |
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