20220113, 03:12  #276  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·3·853 Posts 
Quote:


20220113, 06:31  #277 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
110010000010_{2} 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 Note 2: All GFN base b and all GRU base b are strongprobableprimes (primes and strong pseudoprimes) to base b, since they are overprobableprimes (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 secondlargest 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). 
20220113, 06:36  #278 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,601 Posts 
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 20220113 at 06:37 
20220113, 11:42  #279  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,601 Posts 
Quote:
the original text file for base 35 is too large (1743 KB) to upload, thus zipped it Last fiddled with by sweety439 on 20220113 at 11:43 

20220115, 07:11  #280 
"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 secondlargest 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 secondlargest minimal primes (start with b+1) have the numbers of digits: (combine with the thirdlargest and the fourthlargest 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 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 2digit 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 3digit minimal prime (start with b+1), also all bases b>4 have a 4digit minimal primes (start with b+1), all bases b>4 have a 5digit minimal primes (start with b+1), etc. (note that in the original minimal prime (i.e. prime > base is not required), all singledigit primes are minimal primes, and I conjectured that all bases b != 8 have a 2digit minimal prime, all bases b != 2, 4, 6, 7 have a 3digit minimal prime, all bases b != 2, 3, 4, 5, 7 have a 4digit minimal prime, all bases b != 2, 3, 4, 9 have a 5digit minimal prime, etc. (the bases with no ndigit 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 20220115 at 15:54 
20220117, 06:17  #281 
"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 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
generalized minimal (probable) primes  sweety439  sweety439  136  20211128 19:42 
Minimal set of the strings for primes with at least two digits  sweety439  Miscellaneous Math  6  20191125 07:37 
Digit strings containing primes  davar55  Puzzles  13  20180315 14:46 
Primes from powers of 2 strings.  Flatlander  Puzzles  40  20110210 09:42 
Strings of Digits  davar55  Puzzles  5  20081102 00:08 