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2021-11-28, 17:20   #221
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1100100010002 Posts

These are all minimal primes (start with b+1) in base b=25 up to 2^32

Base 25 is a very hard base (however, of course, bases > 25 which are coprime to 6 are harder than it), we can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of minimal primes (start with b+1) in base b=17 (or 19, 21, 22, 23, 28, 30, 36) (including primality proving of all primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of minimal primes (start with b+1) in base b=25 (or 26, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens.
Attached Files
 base 25.txt (458.2 KB, 40 views)

Last fiddled with by sweety439 on 2021-12-11 at 18:48

2021-11-28, 17:23   #222
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23×401 Posts

Quote:
 Originally Posted by sweety439 Upload text files, searched up to length 5000 For 11{0}1 (the dual of 1{0}11), see https://www.rieselprime.de/ziki/Williams_prime_PP_least For 10{z} (the dual of {z}yz), see https://www.rieselprime.de/ziki/Williams_prime_PM_least
This is the text file for {z0}z1 (i.e. generalized Wagstaff primes, see A084742, but exclude p=3, and use the length of the primes (i.e. use p-1 instead of p)), also searched to length 5000

very important note: they are minimal primes (start with b'+1) in base b'=b^2 instead of base b'=b
Attached Files
 z0z0z1.txt (8.5 KB, 38 views)

Last fiddled with by sweety439 on 2021-11-28 at 17:25

2021-11-30, 19:59   #223
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

320810 Posts

more datas
Attached Files
 base 25.txt (551.0 KB, 33 views) base 20.txt (19.9 KB, 33 views) base 24.txt (19.2 KB, 35 views)

2021-11-30, 20:41   #224
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

62108 Posts

Quote:
 Originally Posted by sweety439 In base 10, the set such strings are not simply to write, however, if "primes > base" is not needed, then such strings are any strings n such that A039995(n) = 0 (not A062115, since A062115 is for substring instead of subsequence, i.e. A062115 is the numbers n such that A039997(n) = 0 instead of the numbers n such that A039995(n) = 0) with any number (including 0) of leading zeros. Such strings are called primefree strings in this post.
In base 10, a sting is primefree string if and only if:

The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms:

* empty string
* single-digit number
* "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0)
* X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7)
* 28{0}7
* 4{6}9
* 221
* 2021
* 2201
* 22001
* 220001
* 2200001
* (5^n)1 with n<11
* 581
* 5(0^n)27 with n<28
* 5207
* 52007
* 520007
* 649
* 6649
* 66649
* 6049
* 60049
* 600049
* 6000049
* 66049
* 660049
* 6600049
* 666049
* 6660049
* 8(5^n)1 with n<11
* 8051
* 80551
* 805551
* 8055551
* 91
* 901
* 921
* 951
* 981
* 9021
* 9051
* 9081
* 9201
* 9501
* 9581
* 9801
* 90581
* 949
* 9469
* 94669

reference: https://math.stackexchange.com/quest...e-number-in-it

Last fiddled with by sweety439 on 2021-12-20 at 21:44 Reason: remove 95081 from the list since 5081 is prime

2021-11-30, 20:52   #225
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23·401 Posts

Update data for minimal primes, see https://sites.google.com/view/data-of-minimal-primes
Attached Files
 data for minimal primes.txt (5.9 KB, 34 views)

2021-12-06, 10:14   #226
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23×401 Posts

Quote:
 Originally Posted by sweety439 Related search for minimal primes (generalized form: (a*b^n+c)/d) in PRP top: b^n+c b^n-c a*b^n+c a*b^n-c (b^n+c)/d (b^n-c)/d (a*b^n+c)/d (a*b^n-c)/d Also for the special case c = +-1 and d = 1, they are proven primes, the search page in top 5000 primes: https://primes.utm.edu/primes/search_proth.php

 2021-12-06, 13:17 #227 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 11111100002 Posts another message _ message transfer to small corner of net Apprciation is felt for your effort @sweety439 and all
2021-12-07, 14:15   #228
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23·401 Posts

Quote:
 Originally Posted by sweety439 The families which are excepted as contain no primes, but undecidable at this point in time, for these 369 bases are: (totally 377 families) * 4:{0}:1, 16:{0}:1 for b = 32 * 12:{62}:63 for b = 125 (Note: {62}:63 for b = 125 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 12:{62}:63 for b = 125 (if exists) must be minimal prime (start with b+1) in base b = 125) * 16:{0}:1 for b = 128 * 36:{0}:1 for b = 216 * 24:{171}:172 for b = 343 (Note: {171}:172 for b = 343 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 24:{171}:172 for b = 343 (if exists) must be minimal prime (start with b+1) in base b = 343) * 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512 * 10:{0}:1, 100:{0}:1 for b = 1000 * 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024 * 1:{0}:1 for other even bases b * {((b-1)/2)}:((b+1)/2) for other odd bases b
For GFN families in bases b <= 1024 which are odd powers (i.e. in https://oeis.org/A070265) (other bases already have information in http://jeppesn.dk/generalized-fermat.html and http://www.noprimeleftbehind.net/crus/GFN-primes.htm and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, which are 1:{0}:1 for even bases and {((b-1)/2)}:((b+1)/2) for odd bases):

b=8:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3

b=27:

{13}:14, ruled out as only contain composite numbers
1:{13}:14, prime at length 2, also 1:{13} has prime at length 3
4:{13}:14, prime at length 11, also 4:{13} has prime at length 24

b=32:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 4
4:{0}:1, unsolved family
8:{0}:1, prime at length 2
16:{0}:1, unsolved family

b=64:

1:{0}:1, ruled out as only contain composite numbers
4:{0}:1, prime at length 2
16:{0}:1, prime at length 3

b=125:

{62}:63, ruled out as only contain composite numbers
2:{62}:63, prime at length 2
12:{62}:63, unsolved family

b=128:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3
8:{0}:1, ruled out as only contain composite numbers
16:{0}:1, unsolved family
32:{0}:1, ruled out as only contain composite numbers
64:{0}:1, ruled out as only contain composite numbers

b=216:

1:{0}:1, ruled out as only contain composite numbers
6:{0}:1, prime at length 2
36:{0}:1, unsolved family

b=243:

{121}:122, ruled out as only contain composite numbers
1:{121}:122, prime at length 4, also 1:{121} has prime at length 15
4:{121}:122, prime at length 7, also 4:{121} has prime at length 2
13:{121}:122, although no known prime or PRP in this family, but 13:{121} has prime at length 3, thus 13:{121}:122 is still not unsolved family since the smallest prime (if exists) in this family will not be minimal prime (start with b+1)
40:{121}:122, prime at length 13, however 40:{121} has no known prime or PRP, and 40:{121} is unsolved family (however, 40:{121} is GRU family instead of GFN family, for reference of this family, see https://oeis.org/A028491, there are no single known number in https://oeis.org/A028491 which is == 4 mod 5)

b=343:

{171}:172, ruled out as only contain composite numbers
3:{171}:172, prime at length 2
24:{171}:172, unsolved family

b=512:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, unsolved family
4:{0}:1, unsolved family
8:{0}:1, ruled out as only contain composite numbers
16:{0}:1, unsolved family
32:{0}:1, unsolved family
64:{0}:1, ruled out as only contain composite numbers
128:{0}:1, prime at length 2
256:{0}:1, unsolved family

b=729:

{364}:365, ruled out as only contain composite numbers
4:{364}:365, prime at length 6
40:{364}:365, prime at length 3

b=1000:

1:{0}:1, ruled out as only contain composite numbers
10:{0}:1, unsolved family
100:{0}:1, unsolved family

b=1024:

1:{0}:1, ruled out as only contain composite numbers
4:{0}:1, unsolved family
16:{0}:1, unsolved family
64:{0}:1, prime at length 2
256:{0}:1, unsolved family

Last fiddled with by sweety439 on 2021-12-11 at 15:34

2021-12-08, 08:21   #229
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1100100010002 Posts

Quote:
 Originally Posted by sweety439 In base 10, a sting is primefree string if and only if: The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms: * empty string * single-digit number * "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0) * X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7) * 28{0}7 * 4{6}9 * 221 * 2021 * 2201 * 22001 * 220001 * 2200001 * (5^n)1 with n<11 * 5(0^n)27 with n<28 * 5207 * 52007 * 520007 * 649 * 6649 * 66649 * 6049 * 60049 * 600049 * 6000049 * 66049 * 660049 * 6600049 * 666049 * 6660049 * 8(5^n)1 with n<11 * 8051 * 80551 * 805551 * 8055551 * 91 * 901 * 921 * 951 * 981 * 9021 * 9051 * 9081 * 9201 * 9501 * 9581 * 9801 * 90581 * 949 * 9469 * 94669
Like minimal element of prime numbers > b in base b, we can find the maximal element of primefree strings:

{0}S{0,2,4,5,6,8} with these sets S:

* {0,3,6,9}
* {0,7}
* 2{0}1
* 5{0}1
* 5{0}7
* 8{0}1
* 28{0}7
* 4{6}9
* 2021
* 2200001
* (5^10)1
* 581
* 5(0^27)27
* 520007
* 6000049
* 6600049
* 6660049
* 8(5^10)1
* 8055551
* 9021
* 9201
* 9801
* 90581
* 94669

(note: {0,2,4,6,8} and {0,5} are already included in the digits after S, i.e. {0,2,4,5,6,8})
(note: 2{0}7 and 8{0}7 are already sub-families of 28{0}7)
(note: 66649 is already subsequence of 6660049)
(note: 581, 9051, 9081, 9501, 9581 are already subsequences of 90581 or/and 95081)

Last fiddled with by sweety439 on 2021-12-20 at 21:45 Reason: remove 95081 from the list since 5081 is prime

 2021-12-08, 08:29 #230 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C8816 Posts and we have these "maximal primefree strings" b=2: * {0}1{0} b=3: * {0}1{0}1{0} * {0,2} b=4: * {0}2{0}1{0,2} * {0}{0,3}{0,2} b=10: * {0}{0,3,6,9}{0,2,4,5,6,8} * {0}{0,7}{0,2,4,5,6,8} * {0}2{0}1{0,2,4,5,6,8} * {0}5{0}1{0,2,4,5,6,8} * {0}5{0}7{0,2,4,5,6,8} * {0}8{0}1{0,2,4,5,6,8} * {0}28{0}7{0,2,4,5,6,8} * {0}4{6}9{0,2,4,5,6,8} * {0}2021{0,2,4,5,6,8} * {0}2200001{0,2,4,5,6,8} * {0}55555555551{0,2,4,5,6,8} * {0}500000000000000000000000000027{0,2,4,5,6,8} * {0}520007{0,2,4,5,6,8} * {0}6000049{0,2,4,5,6,8} * {0}6600049{0,2,4,5,6,8} * {0}6660049{0,2,4,5,6,8} * {0}855555555551{0,2,4,5,6,8} * {0}8055551{0,2,4,5,6,8} * {0}9021{0,2,4,5,6,8} * {0}9201{0,2,4,5,6,8} * {0}9801{0,2,4,5,6,8} * {0}90581{0,2,4,5,6,8} * {0}95081{0,2,4,5,6,8} * {0}94669{0,2,4,5,6,8} Last fiddled with by sweety439 on 2021-12-08 at 14:20
 2021-12-10, 16:59 #231 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C8816 Posts These factor pattern can show that such families contain no primes > base: Reference: the divisibility rule for base b: * For prime p dividing b, the number is divisible by p if and only if the last digit of this number is divisible by p. * For prime p dividing b-1, the number is divisible by p if and only if the sum of the digits of this number is divisible by p. * For prime p dividing b+1, the number is divisible by p if and only if the alternating sum of the digits of this number is divisible by p. (this can also show that all palindromic primes in any base b have an odd number of digits, the only possible exception is "11" in base b) (in these examples, only list the numbers > base) Example 1: base 10, family 4{6}9 Code: 49 = 7 * 7 469 = 7 * 67 4669 = 7 * 667 46669 = 7 * 6667 466669 = 7 * 66667 4666669 = 7 * 666667 Example 2: base 10, family 28{0}7 Code: 287 = 7 * 41 2807 = 7 * 401 28007 = 7 * 4001 280007 = 7 * 40001 2800007 = 7 * 400001 28000007 = 7 * 4000001 Example 3: base 9, family {1} Code: 11 = 2 * 5 111 = 7 * 14 1111 = 22 * 45 11111 = 67 * 144 111111 = 222 * 445 1111111 = 667 * 1444 11111111 = 2222 * 4445 111111111 = 6667 * 14444 1111111111 = 22222 * 44445 11111111111 = 66667 * 144444 111111111111 = 222222 * 444445 1111111111111 = 666667 * 1444444 Example 4: base 9, family 3{8} Code: 38 = 5 * 7 388 = 18 * 21 3888 = 58 * 61 38888 = 188 * 201 388888 = 588 * 601 3888888 = 1888 * 2001 38888888 = 5888 * 6001 388888888 = 18888 * 20001 3888888888 = 58888 * 60001 38888888888 = 188888 * 200001 388888888888 = 588888 * 600001 Example 5: base 8, family 1{0}1 Code: 11 = 3 * 3 101 = 5 * 15 1001 = 11 * 71 10001 = 21 * 361 100001 = 41 * 1741 1000001 = 101 * 7701 10000001 = 201 * 37601 100000001 = 401 * 177401 1000000001 = 1001 * 777001 10000000001 = 2001 * 3776001 100000000001 = 4001 * 17774001 1000000000001 = 10001 * 77770001 Example 6: base 11, family 2{5} Code: 25 = 3 * 9 255 = 2 * 128 2555 = 3 * 919 25555 = 2 * 12828 255555 = 3 * 91919 2555555 = 2 * 1282828 25555555 = 3 * 9191919 255555555 = 2 * 128282828 2555555555 = 3 * 919191919 25555555555 = 2 * 12828282828 Example 7: base 12, family {B}9B Code: 9B = 7 * 15 B9B = 11 * AB BB9B = B7 * 105 BBB9B = 11 * B0AB BBBB9B = BB7 * 1005 BBBBB9B = 11 * B0B0AB BBBBBB9B = BBB7 * 10005 BBBBBBB9B = 11 * B0B0B0AB BBBBBBBB9B = BBBB7 * 100005 BBBBBBBBB9B = 11 * B0B0B0B0AB Example 8: base 14, family B{0}1 Code: B1 = 5 * 23 B01 = 3 * 395 B001 = 5 * 22B3 B0001 = 3 * 39495 B00001 = 5 * 22B2B3 B000001 = 3 * 3949495 B0000001 = 5 * 22B2B2B3 B00000001 = 3 * 394949495 B000000001 = 5 * 22B2B2B2B3 B0000000001 = 3 * 39494949495 Example 9: base 13, family 3{0}95 Code: 395 = 14 * 2B 3095 = 7 * 58A 30095 = 5 * 7A71 300095 = 7 * 5758A 3000095 = 14 * 23A92B 30000095 = 7 * 575758A 300000095 = 5 * 7A527A71 3000000095 = 7 * 57575758A 30000000095 = 14 * 23A923A92B 300000000095 = 7 * 5757575758A 3000000000095 = 5 * 7A527A527A71 Example 10: base 16, family {4}D Code: 4D = 7 * B 44D = 3 * 16F 444D = D * 541 4444D = 7 * 9C0B 44444D = 3 * 16C16F 444444D = D * 540541 4444444D = 7 * 9C09C0B 44444444D = 3 * 16C16C16F 444444444D = D * 540540541 4444444444D = 7 * 9C09C09C0B 44444444444D = 3 * 16C16C16C16F 444444444444D = D * 540540540541 Example 11: base 17, family 1{9} Code: 19 = 2 * D 199 = B * 27 1999 = 2 * D4D 19999 = AB * 287 199999 = 2 * D4D4D 1999999 = AAB * 2887 19999999 = 2 * D4D4D4D 199999999 = AAAB * 28887 1999999999 = 2 * D4D4D4D4D 19999999999 = AAAAB * 288887 199999999999 = 2 * D4D4D4D4D4D 1999999999999 = AAAAAB * 2888887 Example 12: base 36, family O{Z} Code: OZ = T * V OZZ = 4Z * 51 OZZZ = TZ * U1 OZZZZ = 4ZZ * 501 OZZZZZ = TZZ * U01 OZZZZZZ = 4ZZZ * 5001 OZZZZZZZ = TZZZ * U001 OZZZZZZZZ = 4ZZZZ * 50001 OZZZZZZZZZ = TZZZZ * U0001 OZZZZZZZZZZ = 4ZZZZZ * 500001 OZZZZZZZZZZZ = TZZZZZ * U00001 OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001 Some references of this, see: http://www.worldofnumbers.com/wing.htm for: {1}0{1} (base 10): Code: 101 = 1 * 101 (the only possible prime case) 11011 = 11 * 1001 1110111 = 111 * 10001 111101111 = 1111 * 100001 11111011111 = 11111 * 1000001 1111110111111 = 111111 * 10000001 {1}2{1} (base 10): Code: 121 = 11 * 11 11211 = 101 * 111 1112111 = 1001 * 1111 111121111 = 10001 * 11111 11111211111 = 100001 * 111111 1111112111111 = 1000001 * 1111111 {3}2{3} (base 10): Code: 323 = 17 * 19 33233 = 167 * 199 3332333 = 1667 * 1999 333323333 = 16667 * 19999 33333233333 = 166667 * 199999 3333332333333 = 1666667 * 1999999 {3}4{3} (base 10): Code: 343 = 7 * 49 33433 = 67 * 499 3334333 = 667 * 4999 333343333 = 6667 * 49999 33333433333 = 66667 * 499999 3333334333333 = 666667 * 4999999 http://www.worldofnumbers.com/deplat.htm for: 1{2}1 (base 10): Code: 11 = 11 * 1 (the only possible prime case) 121 = 11 * 11 1221 = 11 * 111 12221 = 11 * 1111 122221 = 11 * 11111 1222221 = 11 * 111111 12222221 = 11 * 1111111 7{3}7 (base 10): Code: 77 = 11 * 7 737 = 11 * 67 7337 = 11 * 667 73337 = 11 * 6667 733337 = 11 * 66667 7333337 = 11 * 666667 73333337 = 11 * 6666667 9{7}9 (base 10): Code: 99 = 11 * 9 979 = 11 * 89 9779 = 11 * 889 97779 = 11 * 8889 977779 = 11 * 88889 9777779 = 11 * 888889 97777779 = 11 * 8888889 9{4}9 (base 10): Code: 99 = 11 * 9 949 = 13 * 73 9449 = 11 * 859 94449 = 3 * 31483 944449 = 11 * 85859 9444449 = 7 * 1349207 94444449 = 11 * 8585859 944444449 = 13 * 72649573 9444444449 = 11 * 858585859 94444444449 = 3 * 31481481483 944444444449 = 11 * 85858585859 9444444444449 = 7 * 1349206349207 94444444444449 = 11 * 8585858585859 http://www.worldofnumbers.com/Append...s%20to%20n.txt for: 37{1} (base 10): Code: 37 = 37 * 1 (the only possible prime case) 371 = 7 * 53 3711 = 3 * 1237 37111 = 37 * 1003 371111 = 13 * 28547 3711111 = 3 * 1237037 37111111 = 37 * 1003003 371111111 = 7 * 53015873 3711111111 = 3 * 1237037037 37111111111 = 37 * 1003003003 371111111111 = 13 * 28547008547 3711111111111 = 3 * 1237037037037 37111111111111 = 37 * 1003003003003 38{1} (base 10): Code: 38 = 2 * 19 381 = 3 * 127 3811 = 37 * 103 38111 = 23 * 1657 381111 = 3 * 127037 3811111 = 37 * 103003 38111111 = 233 * 163567 381111111 = 3 * 127037037 3811111111 = 37 * 103003003 38111111111 = 2333 * 16335667 381111111111 = 3 * 127037037037 3811111111111 = 37 * 103003003003 38111111111111 = 23333 * 1633356667 Last fiddled with by sweety439 on 2021-12-19 at 23:20

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