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2022-01-13, 03:12   #276
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

22·3·853 Posts

Quote:
 Originally Posted by sweety439 I either post on this thread or update my article (or both).
Maybe just over on your google doc. Then people can fetch it at their leisure. When you post here, you generate notifications for people that there is something new. While it is new, it is not worth noticing.

2022-01-13, 06:31   #277
sweety439

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

1100100000102 Posts

This is the smallest GFN primes and the smallest GRU primes in bases b<=64

Note 1: we do not include the case where the base of the GFNs is perfect odd power and the case where the base of the GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=64 are:

Code:
base         GFN family         GRU family
4                               {1}
8            1{0}1              {1}
9                               {1}
16                              {1}, 1{5}, {C}D
25                              {1}
27           {D}E               {1}
32           1{0}1              {1}
36                              {1}
49                              {1}
64           1{0}1              {1}, 1{L}, 5{L}, 1{9}, {u}v
Such small primes are: 11 in base 4, 111 in base 8, 11 in base 16, 111 in base 27, 11 in base 36, 19 in base 64

Note 2: All GFN base b and all GRU base b are strong-probable-primes (primes and strong pseudoprimes) to base b, since they are over-probable-primes (primes and overpseudoprimes) to base b (references: https://oeis.org/A141232 http://arxiv.org/abs/0806.3412 http://arxiv.org/abs/0807.2332 http://arxiv.org/abs/1412.5226 https://cs.uwaterloo.ca/journals/JIS...shevelev19.pdf), and all overpseudoprimes are strong pseudoprimes to the same base b, all strong pseudoprimes are Euler–Jacobi pseudoprimes to the same base b, all Euler–Jacobi pseudoprimes are Euler pseudoprimes to the same base b, all Euler pseudoprimes are Fermat pseudoprimes to the same base b, so don't test with this base (see https://mersenneforum.org/showthread.php?t=10476&page=2, https://oeis.org/A171381, https://oeis.org/A028491), note that there are also (but very few) numbers in the simple families which are neither GFN families nor GRU families, which are pseudoprimes, e.g. for the family {5}25 in base 8 (which have the smallest prime 555555555555525, corresponding to the second-largest base 8 minimal prime (start with b+1)), a smaller number 525 is 341 in decimal, which is Fermat pseudoprime and Euler pseudoprime (although not strong pseudoprime, but I think there are many examples of strong pseudoprimes to base 2 and/or base 3) to base 2 (and thus to base 8, since pseudoprimes to base b are always (the same type) pseudoprimes to base b^r for all r>1, and 8=2^3).
Attached Files
 smallest GFN and smallest GRU.txt (17.0 KB, 6 views)

2022-01-13, 06:36   #278
sweety439

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

2×1,601 Posts

Quote:
 Originally Posted by Uncwilly it is not worth noticing.
but it is my newer researching result, and I think that it is important, sometimes I update my old posts, such as #208 and #215, also there are posts which are lists of references: #140 and #154

Last fiddled with by sweety439 on 2022-01-13 at 06:37

2022-01-13, 11:42   #279
sweety439

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

2·1,601 Posts

Quote:
 Originally Posted by sweety439 done to base b=31 the original text file for base 31 is too large (1546 KB) to upload, thus zipped it
all bases b<=36 are done.

the original text file for base 35 is too large (1743 KB) to upload, thus zipped it
Attached Files
 kernel32.txt (539.7 KB, 5 views) kernel33.txt (871.2 KB, 5 views) kernel34.txt (550.9 KB, 5 views) kernel35.zip (694.7 KB, 6 views) kernel36.txt (161.0 KB, 5 views)

Last fiddled with by sweety439 on 2022-01-13 at 11:43

 2022-01-15, 07:11 #280 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×1,601 Posts Conjecture: There is no base b such that the largest minimal prime (start with b+1) and the second-largest minimal prime (start with b+1) have the same number of digits in base b, note that in the original minimal prime (i.e. prime > base is not required), the three largest minimal primes in decimal (base 10) all have the same number of digits (60000049, 66000049, 66600049, all have 8 digits), and in base 2 the largest (and the only) two minimal primes are 10 and 11, both have 2 digits, also, in base 5, the largest two minimal primes are 14444 and 44441, both have 5 digits. For the problem in this project (i.e. the minimal primes (start with b+1)), the largest and the second-largest minimal primes (start with b+1) have the numbers of digits: (combine with the third-largest and the fourth-largest minimal primes (start with b+1), see the table below): Code: base 1st largest 2nd largest 3rd largest 4th largest 2 2 N/A N/A N/A 3 3 2 2 N/A 4 3 2 2 2 5 96 6 5 5 6 5 4 4 2 7 17 10 8 7 8 221 15 13 11 9 1161 689 331 38 (conjectured) 10 31 12 8 8 12 42 30 9 8 and (the number of digits of 1st largest) / (the number of digits of 2nd largest) getting large very quickly if (b-1)*eulerphi(b) gets large, thus I do not think such base can exist. Another conjecture: For any number n>=2, there exists a minimal primes (start with b+1) with exactly n digits in base b, for every enough large b Clearly, all 2-digit primes (except "10" (i.e. = b) when b itself is prime) are minimal primes (start with b+1) base b, I conjectured that all bases b != 2, 6 have a 3-digit minimal prime (start with b+1), also all bases b>4 have a 4-digit minimal primes (start with b+1), all bases b>4 have a 5-digit minimal primes (start with b+1), etc. (note that in the original minimal prime (i.e. prime > base is not required), all single-digit primes are minimal primes, and I conjectured that all bases b != 8 have a 2-digit minimal prime, all bases b != 2, 4, 6, 7 have a 3-digit minimal prime, all bases b != 2, 3, 4, 5, 7 have a 4-digit minimal prime, all bases b != 2, 3, 4, 9 have a 5-digit minimal prime, etc. (the bases with no n-digit minimal prime for given n is more complex, thus the problem in this project (i.e. the minimal primes (start with b+1)) is really better)) (for more data, see post 145) we can research: * the possible length of the minimal primes (start with b+1) * the possible (first,last) combo of the minimal primes (start with b+1) * for these minimal primes (start with b+1), the digit which appears the most times in this minimal prime (start with b+1) * the length such that there are the most minimal primes (start with b+1) * the (first,last) combo such that there are the most minimal primes (start with b+1) Last fiddled with by sweety439 on 2022-01-15 at 15:54
 2022-01-17, 06:17 #281 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·1,601 Posts find the set of the minimal primes (start with b+1) base b for various bases b (2<=b<=36) is the target of the project in this thread. minimal prime (start with b+1) base b is always minimal prime (start with b'+1) base b' = b^n, if it is > b', for any integer n>1 original minimal prime (i.e. prime > b is not required) base b is always minimal prime (start with b+1) base b, if it is > b

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