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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

sweety439 2022-07-17 16:01

[QUOTE=sweety439;609479]If we write all minimal primes (start with b+1) in base b one time, then we will write these numbers of digits:

[CODE]
b=2:

0 0's
2 1's

b=3:

0 0's
5 1's
2 2's

b=4:

0 0's
5 1's
3 2's
3 3's

b=5:

105 0's
24 1's
4 2's
22 3's
14 4's

b=6:

3 0's
9 1's
2 2's
2 3's
8 4's
5 5's

b=7:

43 0's
73 1's
21 2's
67 3's
24 4's
51 5's
9 6's

b=8:

22 0's
49 1's
13 2's
24 3's
283 4's
48 5's
25 6's
59 7's

b=9:

1350 0's
174 1's
32 2's
108 3's
24 4's
107 5's
357 6's
794 7's
58 8's

b=10:

69 0's
33 1's
27 2's
9 3's
19 4's
45 5's
21 6's
31 7's
22 8's
34 9's

b=11:

2666 0's
523 1's
227 2's
250 3's
722 4's
1514 5's (unless 5(7^62668) is in fact composite and there is no prime of the form 5{7}, in this case there are 1513 5's, but this is very impossible)
251 6's
66917 7's (unless 5(7^62668) is in fact composite, in this case there are n+4249 (must be > 66917) 7's where n is the smallest number n such that 5(7^n) is prime (must be > 62668) if there is a prime of the form 5{7}, or there are 4249 7's if there is no prime of the form 5{7}, but 5(7^62668) is in fact composite is very impossible)
357 8's
592 9's
1395 A's

b=12:

105 0's
42 1's
28 2's
4 3's
25 4's
23 5's
14 6's
43 7's
4 8's
38 9's
38 A's
69 B's
[/CODE][/QUOTE]

Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, [URL="http://factordb.com/index.php?id=3"]sum[/URL], [URL="http://factordb.com/index.php?id=3"]product[/URL]

Base 3:

3 primes, totally 7 digits, [URL="http://factordb.com/index.php?id=25"]sum[/URL], [URL="http://factordb.com/index.php?id=455"]product[/URL]

Base 4:

5 primes, totally 11 digits, [URL="http://factordb.com/index.php?id=77"]sum[/URL], [URL="http://factordb.com/index.php?id=205205"]product[/URL]

Base 5:

22 primes, totally 169 digits, [URL="http://factordb.com/index.php?id=1100000003799642708"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457822814"]product[/URL]

Base 6:

11 primes, totally 29 digits, [URL="http://factordb.com/index.php?id=7401"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457821560"]product[/URL]

Base 7:

71 primes, totally 288 digits, [URL="http://factordb.com/index.php?id=116315467894207"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457825324"]product[/URL]

Base 8:

75 primes, totally 523 digits, [URL="http://factordb.com/index.php?id=1100000003799644593"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002371473795"]product[/URL]

Base 9:

151 primes, totally 3004 digits, [URL="http://factordb.com/index.php?id=1100000003799645271"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003450366253"]product[/URL]

Base 10:

77 primes, totally 310 digits, [URL="http://factordb.com/index.php?id=1100000003799645582"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002370859491"]product[/URL]

Base 11:

1068 primes, totally 75414 digits, [URL="http://factordb.com/index.php?id=1100000003799646641"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003583737715"]product[/URL]

Base 12:

106 primes, totally 433 digits, [URL="http://factordb.com/index.php?id=1100000003799647067"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457818232"]product[/URL]

Base 14:

650 primes, totally 25404 digits, [URL="http://factordb.com/index.php?id=1100000003799647609"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003575953976"]product[/URL]

Base 15:

1284 primes, totally 8286 digits, [URL="http://factordb.com/index.php?id=1100000003799647942"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003588261354"]product[/URL]

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.

sweety439 2022-07-20 08:03

Two special cases of minimal primes (start with b+1) base b

* lovely numbers base b: Let d_1 and d_2 be digits in base b such that d_1 + d_2 = b, find the smallest prime of the form d_1*b^n+d_2 with n >= 1, this prime is always minimal prime (start with b+1) base b, this includes the special cases d_1 = 1, d_2 = b-1 (which is b^n+(b-1), see [URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]) and d_1 = b-1, d_2 = 1 (which is (b-1)*b^n+1, see [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL] and [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL] and [URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL])
** such prime is always expected to exist as there cannot be covering congruence nor algebraic factorization (since if so, then d_1, d_2, b will be all r-th powers for an odd r > 1, which is impossible (by Fermat Last Theorem)
* flexible numbers base b: Let d be a divisor (>1) of b-1 (if b-1 is prime, then d can only be b-1 itself), find the smallest prime of the form ((d-1)*b^n+1)/d with n >= 2, this prime is always minimal prime (start with b+1) base b
** for the case d = 2, such prime may not exist and it is widely believed that there are only finitely many such primes for fixed base b, since it is generalized half Fermat prime in base b
** for the case d > 2, such prime are usually expected to exist (as there cannot be covering congruence of this form, but there may be algebraic factorization or combine of covering congruence and algebraic factorization if d-1 is indeed perfect odd power (of the form m^r with odd r > 1) or of the form 4*m^4, and if d-1 is of neither of these two forms, then there must be prime of this form), but the smallest such prime may be large, e.g.

[CODE]
b,d,smallest exponent n
13,12,564
17,8,190
23,11,3762
31,6,1026
43,14,580
70,69,555
[/CODE]

sweety439 2022-07-30 02:10

[QUOTE=sweety439;571731]References of given simple families for the minimal primes (start with b+1) problem in bases 2<=b<=1024:

{1}:

[URL="http://www.users.globalnet.co.uk/~aads/primes.html"]http://www.users.globalnet.co.uk/~aads/primes.html[/URL] (broken link: [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]from wayback machine[/URL])
[URL="http://www.users.globalnet.co.uk/~aads/titans.html"]http://www.users.globalnet.co.uk/~aads/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html"]from wayback machine[/URL])
[URL="http://www.primes.viner-steward.org/andy/titans.html"]http://www.primes.viner-steward.org/andy/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html"]from wayback machine[/URL])
[URL="http://www.phi.redgolpe.com/"]http://www.phi.redgolpe.com/[/URL] (broken link: [URL="https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/"]from wayback machine[/URL])
[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]
[URL="https://oeis.org/A128164/a128164_7.txt"]https://oeis.org/A128164/a128164_7.txt[/URL]
[URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL]
[URL="http://www.mersennewiki.org/index.php/Repunit"]http://www.mersennewiki.org/index.php/Repunit[/URL] (broken link: [URL="https://web.archive.org/web/20180416000002/http://www.mersennewiki.org/index.php/Repunit"]from wayback machine[/URL])
[URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL]
[URL="http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470"]http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470[/URL]
[URL="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906"]https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906[/URL] (archive today cannot automatically return the archive page, if you use archive today, click [URL="https://archive.is/WCvbi"]https://archive.is/WCvbi[/URL])
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://ebisui-hirotaka.com/img/file410.pdf"]http://ebisui-hirotaka.com/img/file410.pdf[/URL]
[URL="https://www.jstor.org/stable/2006470?origin=crossref"]https://www.jstor.org/stable/2006470?origin=crossref[/URL]
[URL="http://www.bitman.name/math/table/379"]http://www.bitman.name/math/table/379[/URL]
[URL="https://oeis.org/A084740"]https://oeis.org/A084740[/URL]
[URL="https://oeis.org/A084738"]https://oeis.org/A084738[/URL] (corresponding primes)
[URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL] (prime bases)
[URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL] (prime bases, corresponding primes)
[URL="https://oeis.org/A128164"]https://oeis.org/A128164[/URL] (length 2 not allowed)
[URL="https://oeis.org/A285642"]https://oeis.org/A285642[/URL] (length 2 not allowed, corresponding primes)

1{0}1:

[URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL]
[URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/index.html"]http://yves.gallot.pagesperso-orange.fr/primes/index.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/stat.html"]http://yves.gallot.pagesperso-orange.fr/primes/stat.html[/URL]
[URL="https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf"]https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf"]https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf"]https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf"]https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf[/URL] (b=2^n)
[URL="https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf"]https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf[/URL] (b=2^n)
[URL="https://arxiv.org/pdf/1605.01371.pdf"]https://arxiv.org/pdf/1605.01371.pdf[/URL] (b=2^n)
[URL="https://oeis.org/A228101"]https://oeis.org/A228101[/URL]
[URL="https://oeis.org/A079706"]https://oeis.org/A079706[/URL]
[URL="https://oeis.org/A084712"]https://oeis.org/A084712[/URL] (corresponding primes)
[URL="https://oeis.org/A123669"]https://oeis.org/A123669[/URL] (length 2 not allowed, corresponding primes)

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1:

[URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL] (2{0}1 in base 512, 4{0}1 in bases 32, 512, 1024, which are not in the first 4 references)
[URL="http://www.prothsearch.com/GFN10.html"]http://www.prothsearch.com/GFN10.html[/URL] (A{0}1 in base 1000, which are not in the first 4 references)
[URL="https://mersenneforum.org/showthread.php?t=6918"]https://mersenneforum.org/showthread.php?t=6918[/URL] (2{0}1)
[URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL] (2{0}1 in bases == 11 mod 12)
[URL="https://oeis.org/A119624"]https://oeis.org/A119624[/URL] (2{0}1)
[URL="https://oeis.org/A253178"]https://oeis.org/A253178[/URL] (2{0}1)
[URL="https://oeis.org/A098872"]https://oeis.org/A098872[/URL] (2{0}1 in bases divisible by 6)
[URL="https://oeis.org/A098877"]https://oeis.org/A098877[/URL] (3{0}1 in bases divisible by 6)
[URL="https://oeis.org/A088782"]https://oeis.org/A088782[/URL] (A{0}1)
[URL="https://oeis.org/A088622"]https://oeis.org/A088622[/URL] (A{0}1, corresponding primes)

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}:

[URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL] (1{z})
[URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL] (1{z})
[URL="https://oeis.org/A119591"]https://oeis.org/A119591[/URL] (1{z})
[URL="https://oeis.org/A098873"]https://oeis.org/A098873[/URL] (1{z} in bases divisible by 6)
[URL="https://oeis.org/A098876"]https://oeis.org/A098876[/URL] (2{z} in bases divisible by 6)

z{0}1:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.prothsearch.com/riesel1a.html"]http://www.prothsearch.com/riesel1a.html[/URL] (base 512)
[URL="http://www.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL]
[URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL] (prime bases)

y{z}:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]https://www.rieselprime.de/ziki/Williams_prime_MM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL]
[URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL]
[URL="http://www.prothsearch.com/riesel2.html"]http://www.prothsearch.com/riesel2.html[/URL] (base 128)
[URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL] (prime bases)

1{0}2:

[URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL]
[URL="https://oeis.org/A084713"]https://oeis.org/A084713[/URL] (corresponding primes)
[URL="https://oeis.org/A138067"]https://oeis.org/A138067[/URL] (length 2 not allowed)

1{0}z:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]
[URL="https://oeis.org/A076846"]https://oeis.org/A076846[/URL] (corresponding primes)
[URL="https://oeis.org/A078178"]https://oeis.org/A078178[/URL] (length 2 not allowed)
[URL="https://oeis.org/A078179"]https://oeis.org/A078179[/URL] (length 2 not allowed, corresponding primes)

{z}1:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/435"]http://www.bitman.name/math/table/435[/URL] (prime bases)
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL]
[URL="https://oeis.org/A343589"]https://oeis.org/A343589[/URL] (corresponding primes)
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL] (prime bases)

11{0}1: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]https://www.rieselprime.de/ziki/Williams_prime_PP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/474"]http://www.bitman.name/math/table/474[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

1{0}11: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A346149"]https://oeis.org/A346149[/URL]
[URL="https://oeis.org/A346154"]https://oeis.org/A346154[/URL] (corresponding primes)

10{z}: (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})

[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_least"]https://www.rieselprime.de/ziki/Williams_prime_PM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_table"]https://www.rieselprime.de/ziki/Williams_prime_PM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/471"]http://www.bitman.name/math/table/471[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

{z}y:

[URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] (length 1 allowed)
[URL="https://oeis.org/A250200"]https://oeis.org/A250200[/URL]
[URL="https://oeis.org/A255707"]https://oeis.org/A255707[/URL] (length 1 allowed)
[URL="https://oeis.org/A084714"]https://oeis.org/A084714[/URL] (length 1 allowed, corresponding primes)
[URL="https://oeis.org/A292201"]https://oeis.org/A292201[/URL] (length 1 allowed, prime bases)

{z}yz: (not minimal prime (start with b+1) if there is smaller prime of the form {z}y)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A178250"]https://oeis.org/A178250[/URL]

{#}$: (for odd base b, # = (b−1)/2, $ = (b+1)/2)

[URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]
[URL="http://www.prothsearch.com/GFN05.html"]http://www.prothsearch.com/GFN05.html[/URL] (base 625)

{z0}z1: (almost cannot be minimal prime (start with b+1), since this is not simple family, but always minimal prime (start with b'+1) in base b'=b^2)

[URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171"]https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf[/URL]
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://www.bitman.name/math/table/488"]http://www.bitman.name/math/table/488[/URL]
[URL="https://oeis.org/A084742"]https://oeis.org/A084742[/URL]
[URL="https://oeis.org/A084741"]https://oeis.org/A084741[/URL] (corresponding primes)
[URL="https://oeis.org/A065507"]https://oeis.org/A065507[/URL] (prime bases)[/QUOTE]

(below, "index" means the index of this prime in the minimal primes (start with b+1) set)

(only list families which [B]must[/B] produce minimal primes (start with b+1))

Family {1}: (not exist for bases in [URL="https://oeis.org/A096059"]https://oeis.org/A096059[/URL])

Base 2: 11, length 2, decimal 3, index 1
Base 3: 111, length 3, decimal 13, index 3
Base 4: 11, length 2, decimal 5, index 1
Base 5: 111, length 3, decimal 31, index 8
Base 6: 11, length 2, decimal 7, index 1
Base 7: 11111, length 5, decimal 2801, index 53
Base 8: 111, length 3, decimal 73, index 16
Base 9: not exist
Base 10: 11, length 2, decimal 11, index 1
Base 11: 11111111111111111, length 17, decimal 50544702849929377, index 975
Base 12: 11, length 2, decimal 13, index 1
Base 13: 11111, length 5, decimal 30941, index 494
Base 14: 111, length 3, decimal 211, index 40
Base 15: 111, length 3, decimal 241, index 43
Base 16: 11, length 2, decimal 17, index 1
Base 17: 111, length 3, decimal 307, index 56
Base 18: 11, length 2, decimal 19, index 1
Base 19: 1111111111111111111, length 19, decimal 109912203092239643840221, index 29382
Base 20: 111, length 3, decimal 421, index 73
Base 21: 111, length 3, decimal 463, index 78
Base 22: 11, length 2, decimal 23, index 1

Family 1{0}1: (not exist for bases == 1 mod 2 and bases in [URL="https://oeis.org/A070265"]https://oeis.org/A070265[/URL])

Base 2: 11, length 2, decimal 3, index 1
Base 3: not exist
Base 4: 11, length 2, decimal 5, index 1
Base 5: not exist
Base 6: 11, length 2, decimal 7, index 1
Base 7: not exist
Base 8: not exist
Base 9: not exist
Base 10: 11, length 2, decimal 11, index 1
Base 11: not exist
Base 12: 11, length 2, decimal 13, index 1
Base 13: not exist
Base 14: 101, length 3, decimal 197, index 39
Base 15: not exist
Base 16: 11, length 2, decimal 17, index 1
Base 17: not exist
Base 18: 11, length 2, decimal 19, index 1
Base 19: not exist
Base 20: 101, length 3, decimal 401, index 71
Base 21: not exist
Base 22: 11, length 2, decimal 23, index 1

Family 2{0}1: (not exist for bases == 1 mod 3)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: 21, length 2, decimal 7, index 2
Base 4: not exist
Base 5: 21, length 2, decimal 11, index 2
Base 6: 21, length 2, decimal 13, index 3
Base 7: not exist
Base 8: 21, length 2, decimal 17, index 3
Base 9: 21, length 2, decimal 19, index 4
Base 10: not exist
Base 11: 21, length 2, decimal 23, index 4
Base 12: 2001, length 4, decimal 3457, index 58
Base 13: not exist
Base 14: 21, length 2, decimal 29, index 4
Base 15: 21, length 2, decimal 31, index 5
Base 16: not exist
Base 17: 200000000000000000000000000000000000000000000001, length 48, decimal 13555929465559461990942712143872578804076607708197374744547, index 10094
Base 18: 21, length 2, decimal 37, index 5
Base 19: not exist
Base 20: 21, length 2, decimal 41, index 5
Base 21: 21, length 2, decimal 43, index 6
Base 22: not exist

Family 1{z}: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 12, length 2, decimal 5, index 1
Base 4: 13, length 2, decimal 7, index 2
Base 5: 14444, length 5, decimal 1249, index 16
Base 6: 15, length 2, decimal 11, index 2
Base 7: 16, length 2, decimal 13, index 2
Base 8: 177, length 3, decimal 127, index 21
Base 9: 18, length 2, decimal 17, index 3
Base 10: 19, length 2, decimal 19, index 4
Base 11: 1AA, length 3, decimal 241, index 37
Base 12: 1B, length 2, decimal 23, index 4
Base 13: 1CC, length 3, decimal 337, index 48
Base 14: 1DDDD, length 5, decimal 76831, index 233
Base 15: 1E, length 2, decimal 29, index 4
Base 16: 1F, length 2, decimal 31, index 5
Base 17: 1GG, length 3, decimal 577, index 86
Base 18: 1HH, length 3, decimal 647, index 66
Base 19: 1I, length 2, decimal 37, index 4
Base 20: 1JJJJJJJJJJ, length 11, decimal 20479999999999, index 3015
Base 21: 1K, length 2, decimal 41, index 5
Base 22: 1L, length 2, decimal 43, index 6

Family 3{0}1: (not exist for bases == 1 mod 2)

Base 2: not interpretable (base 2 has no digit "3")
Base 3: not interpretable (base 3 has no digit "3")
Base 4: 31, length 2, decimal 13, index 4
Base 5: not exist
Base 6: 31, length 2, decimal 19, index 5
Base 7: not exist
Base 8: 301, length 3, decimal 193, index 24
Base 9: not exist
Base 10: 31, length 2, decimal 31, index 7
Base 11: not exist
Base 12: 31, length 2, decimal 37, index 7
Base 13: not exist
Base 14: 31, length 2, decimal 43, index 8
Base 15: not exist
Base 16: 301, length 3, decimal 769, index 69
Base 17: not exist
Base 18: 3001, length 4, decimal 17497, index 195
Base 19: not exist
Base 20: 31, length 2, decimal 61, index 10
Base 21: not exist
Base 22: 31, length 2, decimal 67, index 11

Family 2{z}: (not exist for bases == 1 mod 2)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: not exist
Base 4: 23, length 2, decimal 11, index 3
Base 5: not exist
Base 6: 25, length 2, decimal 17, index 4
Base 7: not exist
Base 8: 27, length 2, decimal 23, index 5
Base 9: not exist
Base 10: 29, length 2, decimal 29, index 6
Base 11: not exist
Base 12: 2BB, length 3, decimal 431, index 34
Base 13: not exist
Base 14: 2D, length 2, decimal 41, index 7
Base 15: not exist
Base 16: 2F, length 2, decimal 47, index 9
Base 17: not exist
Base 18: 2H, length 2, decimal 53, index 9
Base 19: not exist
Base 20: 2J, length 2, decimal 59, index 9
Base 21: not exist
Base 22: 2LL, length 3, decimal 1451, index 127

Family 4{0}1: (not exist for bases == 1 mod 5 and bases == 14 mod 15 and bases which are 4th powers)

Base 2: not interpretable (base 2 has no digit "4")
Base 3: not interpretable (base 3 has no digit "4")
Base 4: not interpretable (base 4 has no digit "4")
Base 5: 401, length 3, decimal 101, index 12
Base 6: not exist
Base 7: 41, length 2, decimal 29, index 6
Base 8: 401, length 3, decimal 257, index 27
Base 9: 41, length 2, decimal 37, index 8
Base 10: 41, length 2, decimal 41, index 9
Base 11: not exist
Base 12: 401, length 3, decimal 577, index 35
Base 13: 41, length 2, decimal 53, index 10
Base 14: not exist
Base 15: 41, length 2, decimal 61, index 12
Base 16: not exist
Base 17: 4000001, length 7, decimal 96550277, index 5138
Base 18: 41, length 2, decimal 73, index 14
Base 19: 4001, length 4, decimal 27437, index 748
Base 20: 401, length 3, decimal 1601, index 120
Base 21: not exist
Base 22: 41, length 2, decimal 89, index 16

Family 3{z}: (not exist for bases == 1 mod 3 and bases == 4 mod 5 and bases which are squares)

Base 2: not interpretable (base 2 has no digit "3")
Base 3: not interpretable (base 3 has no digit "3")
Base 4: not exist
Base 5: 34, length 2, decimal 19, index 5
Base 6: 35, length 2, decimal 23, index 6
Base 7: not exist
Base 8: 37, length 2, decimal 31, index 7
Base 9: not exist
Base 10: not exist
Base 11: 3A, length 2, decimal 43, index 9
Base 12: 3B, length 2, decimal 47, index 10
Base 13: not exist
Base 14: not exist
Base 15: 3E, length 2, decimal 59, index 11
Base 16: not exist
Base 17: 3G, length 2, decimal 67, index 12
Base 18: 3H, length 2, decimal 71, index 13
Base 19: not exist
Base 20: 3J, length 2, decimal 79, index 14
Base 21: 3K, length 2, decimal 83, index 15
Base 22: not exist

Family z{0}1: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 21, length 2, decimal 7, index 2
Base 4: 31, length 2, decimal 13, index 4
Base 5: 401, length 3, decimal 101, index 12
Base 6: 51, length 2, decimal 31, index 8
Base 7: 61, length 2, decimal 43, index 10
Base 8: 701, length 3, decimal 449, index 39
Base 9: 81, length 2, decimal 73, index 17
Base 10: 9001, length 4, decimal 9001, index 56
Base 11: A0000000001, length 11, decimal 259374246011, index 905
Base 12: B001, length 4, decimal 19009, index 84
Base 13: C1, length 2, decimal 157, index 31
Base 14: D01, length 3, decimal 2549, index 120
Base 15: E1, length 2, decimal 211, index 41
Base 16: F1, length 2, decimal 241, index 47
Base 17: G0001, length 5, decimal 1336337, index 3039
Base 18: H1, length 2, decimal 307, index 56
Base 19: I00000000000000000000000000001, length 30, decimal 218336795902605993201009018384568383223, index 30322
Base 20: J00000000000001, length 15, decimal 31129600000000000001, index 3160
Base 21: K1, length 2, decimal 421, index 74
Base 22: L1, length 2, decimal 463, index 82

Family y{z}: ((conjectured) exist in all bases)

Base 2: not interpretable (family should have leading zeros or trailing zeros)
Base 3: 12, length 2, decimal 5, index 1
Base 4: 23, length 2, decimal 11, index 3
Base 5: 34, length 2, decimal 19, index 5
Base 6: 45, length 2, decimal 29, index 7
Base 7: 56, length 2, decimal 41, index 9
Base 8: 6777, length 4, decimal 3583, index 55
Base 9: 78, length 2, decimal 71, index 16
Base 10: 89, length 2, decimal 89, index 20
Base 11: 9A, length 2, decimal 109, index 24
Base 12: AB, length 2, decimal 131, index 27
Base 13: BCC, length 3, decimal 2027, index 176
Base 14: CD, length 2, decimal 181, index 36
Base 15: DEEEEEEEEEEEEEE, length 15, decimal 408700964355468749, index 1252
Base 16: EF, length 2, decimal 239, index 46
Base 17: FG, length 2, decimal 271, index 51
Base 18: GHH, length 3, decimal 5507, index 178
Base 19: HIIIIII, length 7, decimal 846825857, index 17286
Base 20: IJ, length 2, decimal 379, index 67
Base 21: JK, length 2, decimal 419, index 73
Base 22: KL, length 2, decimal 461, index 81

Family 1{0}2: (not exist for bases == 0 mod 2 and bases == 1 mod 3)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: 12, length 2, decimal 5, index 1
Base 4: not exist
Base 5: 12, length 2, decimal 7, index 1
Base 6: not exist
Base 7: not exist
Base 8: not exist
Base 9: 12, length 2, decimal 11, index 1
Base 10: not exist
Base 11: 12, length 2, decimal 13, index 1
Base 12: not exist
Base 13: not exist
Base 14: not exist
Base 15: 12, length 2, decimal 17, index 1
Base 16: not exist
Base 17: 12, length 2, decimal 19, index 1
Base 18: not exist
Base 19: not exist
Base 20: not exist
Base 21: 12, length 2, decimal 23, index 1
Base 22: not exist

Family 1{0}z: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 12, length 2, decimal 5, index 1
Base 4: 13, length 2, decimal 7, index 2
Base 5: 104, length 3, decimal 29, index 7
Base 6: 15, length 2, decimal 11, index 2
Base 7: 16, length 2, decimal 13, index 2
Base 8: 107, length 3, decimal 71, index 15
Base 9: 18, length 2, decimal 17, index 3
Base 10: 19, length 2, decimal 19, index 4
Base 11: 10A, length 3, decimal 131, index 26
Base 12: 1B, length 2, decimal 23, index 4
Base 13: 10C, length 3, decimal 181, index 34
Base 14: 1000000000000000D, length 17, decimal 2177953337809371149, index 606
Base 15: 1E, length 2, decimal 29, index 4
Base 16: 1F, length 2, decimal 31, index 5
Base 17: 1000G, length 5, decimal 83537, index 1348
Base 18: 100H, length 4, decimal 5849, index 185
Base 19: 1I, length 2, decimal 37, index 4
Base 20: 10J, length 3, decimal 419, index 72
Base 21: 1K, length 2, decimal 41, index 5
Base 22: 1L, length 2, decimal 43, index 6

Family {z}1: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 21, length 2, decimal 7, index 2
Base 4: 31, length 2, decimal 13, index 4
Base 5: 44441, length 5, decimal 3121, index 20
Base 6: 51, length 2, decimal 31, index 8
Base 7: 61, length 2, decimal 43, index 10
Base 8: 7777777777771, length 13, decimal 549755813881, index 73
Base 9: 81, length 2, decimal 73, index 17
Base 10: 991, length 3, decimal 991, index 44
Base 11: AA1, length 3, decimal 1321, index 111
Base 12: BBBB1, length 5, decimal 248821, index 97
Base 13: C1, length 2, decimal 157, index 31
Base 14: DD1, length 3, decimal 2731, index 131
Base 15: E1, length 2, decimal 211, index 41
Base 16: F1, length 2, decimal 241, index 47
Base 17: GGGGGGGGGG1, length 11, decimal 34271896307617, index 8834
Base 18: H1, length 2, decimal 307, index 56
Base 19: II1, length 3, decimal 6841, index 496
Base 20: JJJJJJJJJJJJJJJJ1, length 17, decimal 13107199999999999999981, index 3185
Base 21: K1, length 2, decimal 421, index 74
Base 22: L1, length 2, decimal 463, index 82

Family {z}y: (not exist for bases == 0 mod 2)

Base 2: not interpretable (family should have leading zeros or trailing zeros)
Base 3: 21, length 2, decimal 7, index 2
Base 4: not exist
Base 5: 43, length 2, decimal 23, index 6
Base 6: not exist
Base 7: 65, length 2, decimal 47, index 11
Base 8: not exist
Base 9: 87, length 2, decimal 79, index 18
Base 10: not exist
Base 11: AAA9, length 4, decimal 14639, index 227
Base 12: not exist
Base 13: CB, length 2, decimal 167, index 33
Base 14: not exist
Base 15: ED, length 2, decimal 223, index 42
Base 16: not exist
Base 17: GGGGGF, length 6, decimal 24137567, index 4999
Base 18: not exist
Base 19: IH, length 2, decimal 359, index 64
Base 20: not exist
Base 21: KJ, length 2, decimal 439, index 77
Base 22: not exist

sweety439 2022-07-30 02:38

[QUOTE=sweety439;571906]The largest possible appearance for given digit d in minimal prime (start with b+1) in base b:

If base b has repunit primes, then the largest possible appearance for digit d=1 in minimal prime (start with b+1) in base b is the length of smallest repunit prime base b (i.e. [URL="https://oeis.org/A084740"]A084740[/URL](b)), the first bases which do not have repunit primes are 9, 25, 32, 49, 64, ...

[CODE]
b=2, d=0: 0
b=2, d=1: 2 (the prime 11)
b=3, d=0: 0
b=3, d=1: 3 (the prime 111)
b=3, d=2: 1 (the primes 12 and 21)
b=4, d=0: 0
b=4, d=1: 2 (the prime 11)
b=4, d=2: 2 (the prime 221)
b=4, d=3: 1 (the primes 13, 23, 31)
b=5, d=0: 93 (the prime 10[SUB]93[/SUB]13)
b=5, d=1: 3 (the prime 111)
b=5, d=2: 1 (the primes 12, 21, 23, 32)
b=5, d=3: 4 (the prime 33331)
b=5, d=4: 4 (the primes 14444 and 44441)
b=6, d=0: 2 (the prime 40041)
b=6, d=1: 2 (the prime 11)
b=6, d=2: 1 (the primes 21 and 25)
b=6, d=3: 1 (the primes 31 and 35)
b=6, d=4: 3 (the prime 4441)
b=6, d=5: 1 (the primes 15, 25, 35, 45, 51)
b=7, d=0: 7 (the prime 5100000001)
b=7, d=1: 5 (the prime 11111)
b=7, d=2: 3 (the prime 1222)
b=7, d=3: 16 (the prime 3[SUB]16[/SUB]1)
b=7, d=4: 2 (the primes 344, 445, 544, 4504, 40054)
b=7, d=5: 4 (the prime 35555)
b=7, d=6: 2 (the prime 6634)
b=8, d=0: 3 (the prime 500025)
b=8, d=1: 3 (the prime 111)
b=8, d=2: 2 (the prime 225)
b=8, d=3: 3 (the prime 3331)
b=8, d=4: 220 (the prime 4[SUB]220[/SUB]7)
b=8, d=5: 14 (the prime 5[SUB]13[/SUB]25)
b=8, d=6: 2 (the primes 661 and 667)
b=8, d=7: 12 (the prime 7[SUB]12[/SUB]1)
b=9, d=0: 1158 (the prime 30[SUB]1158[/SUB]11)
b=9, d=1: 36 (the prime 561[SUB]36[/SUB])
b=9, d=2: 4 (the prime 22227)
b=9, d=3: 8 (the prime 8333333335)
b=9, d=4: 11 (the prime 54[SUB]11[/SUB])
b=9, d=5: 4 (the prime 55551)
b=9, d=6: 329 (the prime 76[SUB]329[/SUB]2)
b=9, d=7: 687 (the prime 27[SUB]686[/SUB]07)
b=9, d=8: 19 (the prime 8[SUB]19[/SUB]335)
b=10, d=0: 28 (the prime 50[SUB]28[/SUB]27)
b=10, d=1: 2 (the prime 11)
b=10, d=2: 3 (the prime 2221)
b=10, d=3: 1 (the primes 13, 23, 31, 37, 43, 53, 73, 83, 349)
b=10, d=4: 2 (the prime 449)
b=10, d=5: 11 (the prime 5[SUB]11[/SUB]1)
b=10, d=6: 4 (the prime 666649)
b=10, d=7: 2 (the primes 277, 577, 727, 757, 787, 877)
b=10, d=8: 2 (the prime 881)
b=10, d=9: 3 (the prime 9949)
b=11, d=0: 126 (the prime 50[SUB]126[/SUB]57)
b=11, d=1: 17 (the prime 1[SUB]17[/SUB])
b=11, d=2: 6 (the prime 5222222)
b=11, d=3: 10 (the prime 3[SUB]10[/SUB]7)
b=11, d=4: 44 (the prime 4[SUB]44[/SUB]1)
b=11, d=5: 221 (the prime 85[SUB]220[/SUB]05]
b=11, d=6: 124 (the prime 326[SUB]124[/SUB])
b=11, d=7: 62668 (the prime 57[SUB]62668[/SUB])
b=11, d=8: 17 (the prime 8[SUB]17[/SUB]3)
b=11, d=9: 32 (the prime 9[SUB]32[/SUB]1)
b=11, d=A: 713 (the prime A[SUB]713[/SUB]58)
b=12, d=0: 39 (the prime 40[SUB]39[/SUB]77)
b=12, d=1: 2 (the prime 11)
b=12, d=2: 3 (the prime 222B)
b=12, d=3: 1 (the primes 31, 35, 37, 3B)
b=12, d=4: 3 (the prime 4441)
b=12, d=5: 2 (the primes 565 and 655)
b=12, d=6: 2 (the prime 665)
b=12, d=7: 3 (the primes 4777 and 9777)
b=12, d=8: 1 (the primes 81, 85, 87, 8B)
b=12, d=9: 4 (the prime 9999B)
b=12, d=A: 4 (the prime AAAA1)
b=12, d=B: 7 (the prime BBBBBB99B)
b=13, d=0: 32017 (the prime 80[SUB]32017[/SUB]111)
b=13, d=1: 5 (the prime 11111)
b=13, d=2: 77 (the prime 72[SUB]77[/SUB])
b=13, d=3: >82000 (the prime A3[SUB]n[/SUB]A)
b=13, d=4: 14 (the prime 94[SUB]14[/SUB])
b=13, d=5: >88000 (the prime 95[SUB]n[/SUB])
b=13, d=6: 137 (the prime 6[SUB]137[/SUB]A3)
b=13, d=7: 1504 (the prime 7[SUB]1504[/SUB]1)
b=13, d=8: 53 (the prime 8[SUB]53[/SUB]7)
b=13, d=9: 1362 (the prime 9[SUB]1362[/SUB]5)
b=13, d=A: 95 (the prime C5A[SUB]95[/SUB])
b=13, d=B: 834 (the prime B[SUB]834[/SUB]74)
b=13, d=C: 10631 (the prime C[SUB]10631[/SUB]92)
b=14, d=0: 83 (the prime 40[SUB]83[/SUB]49)
b=14, d=1: 3 (the prime 111)
b=14, d=2: 3 (the prime B2225)
b=14, d=3: 5 (the prime A33333)
b=14, d=4: 63 (the prime 4[SUB]63[/SUB]09)
b=14, d=5: 36 (the prime 85[SUB]36[/SUB])
b=14, d=6: 10 (the prime 86[SUB]10[/SUB]99)
b=14, d=7: 2 (the primes 771, 77D)
b=14, d=8: 86 (the prime 8[SUB]86[/SUB]B)
b=14, d=9: 37 (the prime 9[SUB]36[/SUB]89)
b=14, d=A: 59 (the prime A[SUB]59[/SUB]3)
b=14, d=B: 78 (the prime 6B[SUB]77[/SUB]2B)
b=14, d=C: 79 (the prime 8C[SUB]79[/SUB]3)
b=14, d=D: 19698 (the prime 4D[SUB]19698[/SUB])
b=15, d=0: 33 (the prime 50[SUB]33[/SUB]17)
b=15, d=1: 3 (the prime 111)
b=15, d=2: 9 (the prime 2222222252)
b=15, d=3: 12 (the prime 3[SUB]12[/SUB]1)
b=15, d=4: 3 (the prime 4434)
b=15, d=5: 8 (the prime 555555557)
b=15, d=6: 104 (the prime 96[SUB]104[/SUB]08)
b=15, d=7: 156 (the prime 7[SUB]155[/SUB]97)
b=15, d=8: 8 (the prime 8888888834)
b=15, d=9: 10 (the prime 9999999999D)
b=15, d=A: 4 (the prime AAAA52)
b=15, d=B: 31 (the prime EB[SUB]31[/SUB])
b=15, d=C: 10 (the prime DCCCCCCCCCC8)
b=15, d=D: 16 (the prime D[SUB]16[/SUB]B)
b=15, d=E: 145 (the prime E[SUB]145[/SUB]397)
b=16, d=0: 3542 (the prime 90[SUB]3542[/SUB]91)
b=16, d=1: 2 (the prime 11)
b=16, d=2: 32 (the prime 2[SUB]32[/SUB]7)
b=16, d=3: >76000 (the prime 3[SUB]n[/SUB]AF)
b=16, d=4: 72785 (the prime 4[SUB]72785[/SUB]DD)
b=16, d=5: 70 (the prime A015[SUB]70[/SUB])
b=16, d=6: 87 (the prime 56[SUB]87[/SUB]F)
b=16, d=7: 20 (the prime 7[SUB]19[/SUB]87)
b=16, d=8: 1517 (the prime F8[SUB]1517[/SUB]F)
b=16, d=9: 1052 (the prime D9[SUB]1052[/SUB])
b=16, d=A: 305 (the prime DA[SUB]305[/SUB]5)
b=16, d=B: 32234 (the prime DB[SUB]32234[/SUB])
b=16, d=C: 3700 (the prime 5BC[SUB]3700[/SUB]D)
b=16, d=D: 39 (the prime 4D[SUB]39[/SUB])
b=16, d=E: 34 (the prime E[SUB]34[/SUB]B)
b=16, d=F: 1961 (the prime 300F[SUB]1960[/SUB]AF)
[/CODE][/QUOTE]

The longest (not only the largest) minimal prime (start with b+1) with given first digit [I]or[/I] given last digit: (of course, first digit must not be 0, and last digit must be coprime to the base (b))

Base 2:

start with 1: 11 (length 2)
end with 1: 11 (length 2)

Base 3:

start with 1: 111 (length 3)
start with 2: 21 (length 2)
end with 1: 111 (length 3)
end with 2: 12 (length 2)

Base 4:

start with 1: 11, 13 (length 2)
start with 2: 221 (length 3)
start with 3: 31 (length 2)
end with 1: 221 (length 3)
end with 3: 13, 23 (length 2)

Base 5:

start with 1: 10[SUB]93[/SUB]13 (length 96)
start with 2: 21, 23 (length 2)
start with 3: 300031 (length 6)
start with 4: 44441 (length 5)
end with 1: 300031 (length 6)
end with 2: 12, 32 (length 2)
end with 3: 10[SUB]93[/SUB]13 (length 96)
end with 4: 14444 (length 5)

Base 6:

start with 1: 11, 15 (length 2)
start with 2: 21, 25 (length 2)
start with 3: 31, 35 (length 2)
start with 4: 40041 (length 5)
start with 5: 51 (length 2)
end with 1: 40041 (length 5)
end with 5: 15, 25, 35, 45 (length 2)

Base 7:

start with 1: 1100021 (length 7)
start with 2: 2111 (length 4)
start with 3: 33333333333333331 (length 17)
start with 4: 40054 (length 5)
start with 5: 5100000001 (length 10)
start with 6: 6034, 6634 (length 4)
end with 1: 33333333333333331 (length 17)
end with 2: 1022, 1112, 1202, 1222 (length 4)
end with 3: 300053 (length 6)
end with 4: 40054 (length 5)
end with 5: 35555 (length 5)
end with 6: 346 (length 3)

Base 8:

start with 1: 107, 111, 117, 141, 147, 161, 177 (length 3)
start with 2: 225, 255 (length 3)
start with 3: 3344441 (length 7)
start with 4: 4[SUB]220[/SUB]7 (length 221)
start with 5: 555555555555525 (length 15)
start with 6: 60171, 60411, 60741 (length 5)
start with 7: 7777777777771 (length 13)
end with 1: 7777777777771 (length 13)
end with 3: 4043, 4443 (length 4)
end with 5: 555555555555525 (length 15)
end with 7: 4[SUB]220[/SUB]7 (length 221)

Base 9:

start with 1: 1000000000000000000000000057 (length 28)
start with 2: 27[SUB]686[/SUB]07 (length 689)
start with 3: 30[SUB]1158[/SUB]11 (length 1161)
start with 4: 438 (length 3)
start with 5: 56111111111111111111111111111111111111 (length 38)
start with 6: 631111 (length 6)
start with 7: 76[SUB]329[/SUB]2 (length 331)
start with 8: 8888888888888888888335 (length 22)
end with 1: 30[SUB]1158[/SUB]11 (length 1161)
end with 2: 76[SUB]329[/SUB]2 (length 331)
end with 4: 544444444444 (length 12)
end with 5: 8888888888888888888335 (length 22)
end with 7: 27[SUB]686[/SUB]07 (length 689)
end with 8: 33388 (length 5)

Base 10:

start with 1: 11, 13, 17, 19 (length 2)
start with 2: 22000001 (length 8)
start with 3: 349 (length 3)
start with 4: 409, 449, 499 (length 3)
start with 5: 5000000000000000000000000000027 (length 31)
start with 6: 60000049, 66000049, 66600049 (length 8)
start with 7: 727, 757, 787 (length 3)
start with 8: 80555551 (length 8)
start with 9: 946669 (length 6)
end with 1: 555555555551 (length 12)
end with 3: 13, 23, 43, 53, 73, 83 (length 2)
end with 7: 5000000000000000000000000000027 (length 31)
end with 9: 60000049, 66000049, 66600049 (length 8)

Base 11:

start with 1: 10[SUB]125[/SUB]51 (length 128)
start with 2: 2888882883, 2888888883 (length 10)
start with 3: 326[SUB]122[/SUB] (length 124)
start with 4: 44777777777777777777777777777777777777777777777777777777777777777 (length 65)
start with 5: 57[SUB]62668[/SUB] (length 62669)
start with 6: 6000000000000083 (length 16)
start with 7: 7[SUB]759[/SUB]44 (length 761)
start with 8: 85[SUB]220[/SUB]05 (length 223)
start with 9: 99777777777777777777777777777777777777777777777777777777777777777 (length 65)
start with A: A[SUB]713[/SUB]58 (length 715)
end with 1: 10[SUB]125[/SUB]51 (length 128)
end with 2: 5555555555555555555555A52 (length 25)
end with 3: 5[SUB]119[/SUB]053 (length 123)
end with 4: 7[SUB]759[/SUB]44 (length 761)
end with 5: 85[SUB]220[/SUB]05 (length 223)
end with 6: 326[SUB]122[/SUB] (length 124)
end with 7: 57[SUB]62668[/SUB] (length 62669)
end with 8: A[SUB]713[/SUB]58 (length 715)
end with 9: 90000000000000000000000000000000000000009799 (length 44)
end with A: 5[SUB]161[/SUB]2A (length 163)

Base 12:

start with 1: 11, 15, 17, 1B (length 2)
start with 2: 2001, 200B, 202B, 222B, 229B, 292B, 299B (length 4)
start with 3: 31, 35, 37, 3B (length 2)
start with 4: 400000000000000000000000000000000000000077 (length 42)
start with 5: 565 (length 3)
start with 6: 600A5 (length 5)
start with 7: 7999B (length 5)
start with 8: 81, 85, 87, 8B (length 2)
start with 9: 9999B (length 5)
start with A: AA000001 (length 8)
start with B: B0000000000000000000000000009B (length 30)
end with 1: AA000001 (length 8)
end with 5: A00065 (length 6)
end with 7: 400000000000000000000000000000000000000077 (length 42)
end with B: B0000000000000000000000000009B (length 30)

sweety439 2022-07-30 02:45

1 Attachment(s)
upload data files (zipped), for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 30, 36

after unzip them, you need to rename "kernel b" to "kernel b.txt" and rename "left b" to "left b.txt", "kernel b" is the data for all known minimal primes (start with b+1) in base b, and "left b" is the data for all unsolved families in base b

search limits for the unsolved families:

base 13 family 9{5} at length 115000
base 13 family A{3}A at length 111000
base 16 family {3}AF at length 98000
all bases 17, 21, 36 families at length 20000

sweety439 2022-08-02 07:49

Just let you know, I know exactly what bases 2 <= b <= 1024 have these families unsolved: (at length 100000) (also exactly what bases 2 <= b <= 1024 have these families proven as only contain composites (only count the numbers > base (b)), by covering congruence, algebraic factorization, or combine of them)

{1}
1{0}1
2{0}1
3{0}1
4{0}1
5{0}1
6{0}1
7{0}1
8{0}1
9{0}1
A{0}1
B{0}1
C{0}1
1{z}
2{z}
3{z}
4{z}
5{z}
6{z}
7{z}
8{z}
9{z}
A{z}
B{z}
z{0}1
y{z}

Also these families, but only at length 20000:

1{0}2
1{0}3
1{0}4
{z}w
{z}x
{z}y
1{0}z
{z}1
{y}z

sweety439 2022-08-10 18:56

[QUOTE=sweety439;609890]* flexible numbers base b: Let d be a divisor (>1) of b-1 (if b-1 is prime, then d can only be b-1 itself), find the smallest prime of the form ((d-1)*b^n+1)/d with n >= 2, this prime is always minimal prime (start with b+1) base b
** for the case d = 2, such prime may not exist and it is widely believed that there are only finitely many such primes for fixed base b, since it is generalized half Fermat prime in base b
** for the case d > 2, such prime are usually expected to exist (as there cannot be covering congruence of this form, but there may be algebraic factorization or combine of covering congruence and algebraic factorization if d-1 is indeed perfect odd power (of the form m^r with odd r > 1) or of the form 4*m^4, and if d-1 is of neither of these two forms, then there must be prime of this form), but the smallest such prime may be large, e.g.

[CODE]
b,d,smallest exponent n
13,12,564
17,8,190
23,11,3762
31,6,1026
43,14,580
70,69,555
[/CODE][/QUOTE]

For the smallest flexible primes base b with given d dividing b-1:

[CODE]
4,3,2
5,4,2
6,5,2
7,3,3
7,6,2
8,7,3
9,4,2
9,8,2
10,3,2
10,9,2
11,5,2
11,10,2
12,11,2
13,3,2
13,4,2
13,6,3
13,12,564
14,13,2
15,7,2
15,14,10
16,3,3
16,5,(not exist, Aurifeuillian factorization of x^4+4*y^4)
16,15,2
17,4,9
17,8,190
17,16,2
18,17,4
19,3,2
19,6,78
19,9,13
19,18,14
20,19,2
21,4,2
21,5,2
21,10,2
21,20,2
22,3,6
22,7,3
22,21,2
23,11,3762
23,22,8
24,23,4
25,3,4
25,4,3
25,6,2
25,8,2
25,12,3
25,24,2
26,5,2
26,25,5
27,13,2
27,26,2
28,3,2
28,9,4
28,27,3
29,4,2
29,7,4
29,14,6
29,28,2
30,29,6
31,3,2
31,5,2
31,6,1026
31,10,24
31,15,99
31,30,2
32,31,2
33,4,3
33,8,2
33,16,2
33,32,252
34,3,3
34,11,2
34,33,3
35,17,2
35,34,20
36,5,45
36,7,5
36,35,2
37,3,3
37,4,6
37,6,4
37,9,2
37,12,4
37,18,12
37,36,6
38,37,4
39,19,3
39,38,2
40,3,3
40,13,3
40,39,2
41,4,3
41,5,6
41,8,2
41,10,15
41,20,2
41,40,4
42,41,2
43,3,12
43,6,38
43,7,4
43,14,580
43,21,3
43,42,24
44,43,3
45,4,28
45,11,5
45,22,2
45,44,2
46,3,3
46,5,2
46,9,7
46,15,3
46,45,2
47,23,2
47,46,2
48,47,4
49,3,2
49,4,2
49,6,3
49,8,8
49,12,26
49,16,2
49,24,4
49,48,2
50,7,2
50,49,3
51,5,2
51,10,2
51,25,5
51,50,2
52,3,3
52,17,5
52,51,17
53,4,4
53,13,2
53,26,4
53,52,24
54,53,2
55,3,2
55,6,2
55,9,2
55,18,2
55,27,6
55,54,2
56,5,78
56,11,2
56,55,2
57,4,2
57,7,5
57,8,2
57,14,6
57,28,44
57,56,2
58,3,2
58,19,2
58,57,3
59,29,2
59,58,4
60,59,2
61,3,10
61,4,2
61,5,6
61,6,3
61,10,4
61,12,6
61,15,11
61,20,70
61,30,4
61,60,2
62,61,4
63,31,3
63,62,4
64,3,2
64,7,2
64,9,(not exist, sum-of-two-cubes factorization)
64,21,24
64,63,11
65,4,2
65,8,2
65,16,5
65,32,2
65,64,2
66,5,15
66,13,2
66,65,2
67,3,6
67,6,6
67,11,19
67,22,(>10000)
67,33,3
67,66,2
68,67,5
69,4,2
69,17,2
69,34,2
69,68,2
70,3,4
70,23,11
70,69,555
71,5,22
71,7,3
71,10,836
71,14,14
71,35,6
71,70,2
72,71,5
73,3,4
73,4,4
73,6,2
73,8,2
73,9,28
73,12,4
73,18,9
73,24,2
73,36,85
73,72,8
74,73,10
75,37,24
75,74,12
76,3,2
76,5,2
76,15,3
76,25,3
76,75,3
77,4,2
77,19,6
77,38,4
77,76,2
78,7,8
78,11,2
78,77,3
79,3,61
79,6,162
79,13,4
79,26,8
79,39,213
79,78,6
80,79,24
81,4,3
81,5,(not exist, Aurifeuillian factorization of x^4+4*y^4)
81,8,2
81,10,29
81,16,2
81,20,5
81,40,2
81,80,4
82,3,2
82,9,10
82,27,3
82,81,6
83,41,104
83,82,680
84,83,2
85,3,2
85,4,2
85,6,13
85,7,4
85,12,6
85,14,2
85,21,57
85,28,2
85,42,10
85,84,6
86,5,6
86,17,2
86,85,2
87,43,2
87,86,2
88,3,28
88,29,2
88,87,3
89,4,6
89,8,6
89,11,288
89,22,2
89,44,2
89,88,132
90,89,2
91,3,2
91,5,9
91,6,4
91,9,5
91,10,36
91,15,4
91,18,4
91,30,8
91,45,16
91,90,140
92,7,51
92,13,4
92,91,4
93,4,156
93,23,2
93,46,2
93,92,4
94,3,51
94,31,12
94,93,2
95,47,8
95,94,2
96,5,3
96,19,2
96,95,84
97,3,7
97,4,2
97,6,2
97,8,2
97,12,???
97,16,2
97,24,9
97,32,4
97,48,28
97,96,2
98,97,137
99,7,6
99,14,6
99,49,2
99,98,6
100,3,3
100,9,7
100,11,2
100,33,2
100,99,5
[/CODE]

sweety439 2022-08-13 13:13

5 Attachment(s)
Results for bases b>16

Base 17 is searched to length 32100
Base 18 is proven (including the primality of the primes)
Base 19 is searched to length 20000

sweety439 2022-08-13 13:16

5 Attachment(s)
Base 20 is proven (including the primality of the primes)
Base 21 is searched to length 20000
Base 22 is proven except the primality of the large strong probable prime B(K^22001)5
Base 23 is now reserved
Base 24 is proven (including the primality of the primes)

sweety439 2022-08-13 13:19

3 Attachment(s)
Base 30 is proven except the primality of the large strong probable prime I(0^24608)D
Base 36 is searched to length 20000

sweety439 2022-08-13 17:17

The base 19 unsolved family 5{H}5 is very low [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]) but eventually should yield a prime (see [URL="http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show[/URL])

Its formula is (107*19^(n+1)-233)/18

(since neither 107 nor 233 is perfect power ([URL="https://oeis.org/A001597"]https://oeis.org/A001597[/URL]), this family has no algebraic factors)

* n == 0 mod 2: factor of 2 (also factor of 4, 5, 10, 20)

this only left n == 1 mod 2

* n == 1 mod 3: factor of 3

this only left n == 3, 5 mod 6

* n == 5 mod 6: factor of 7

this only left n == 3 mod 6

* n == 9 mod 12: factor of 13

this only left n == 3 mod 12

* n == 3 mod 8: factor of 17

this only left n == 15 mod 24

the n = 15 number is divisible by many small primes (11, 29, 47, 71), but all of these primes (p) have large and non-smooth order (znorder(Mod(19,p))), the n = 39 number is divisible by 281, and the n = 63 and 87 numbers have no small prime factors, the n = 111 number is divisible by 89

Base 26 is fully searched to length 20000, and there are 25250 known minimal (probable) primes (start with b+1) and 9 unsolved families, and base 28 is [URL="https://mersenneforum.org/showpost.php?p=546078&postcount=3659"]technically[/URL] fully searched to length 543203 (if we allow probable primes in place of proven primes, see [URL="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf"]http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf[/URL] (see section 3: Recent Results Establishing the Mixed Sierpinski Theorem) and [URL="https://oeis.org/history?seq=A004023&start=50"]https://oeis.org/history?seq=A004023&start=50[/URL] (see M. F. Hasler's discussion in the pink box)), and there are 25528 known minimal (probable) primes (start with b+1), and the only one unsolved family is O{A}F (see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]https://github.com/xayahrainie4793/quasi-mepn-data[/URL] and [URL="https://github.com/curtisbright/mepn-data/blob/master/data/sieve.28.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/sieve.28.txt[/URL]), interestingly, in base 28 there are only 3 known minimal primes (start with b+1) (and it is likely totally 4 minimal primes (start with b+1)) with length > 5271: N(6^24051)LR, 5O(A^31238)F, O4(O^94535)9, and more interestingly, base 28 seems to be high weight base, but there are many families (which [I]must[/I] be minimal primes (start with b+1) in all bases b, if these families are interpretable in this base b) whose smallest length (to make the number prime) (see [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]) set records: 3*b^n+1 (3{0}1, length 8, index 19858), b^n-3 ({z}x, which is {R}P in base 28, length 10, index 23827), b^n-5 ({z}v, which is {R}N in base 28, length 60, index 25401), (b/2)*b^n-1 (#{z}, which is D{R} in base 28, length 48, index 25367), {2}1 (length 40, index 25337)


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