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[QUOTE=sweety439;531552]A (probable) prime was found:
(13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime"[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) currently at n=21939, no (probable) prime found. (4428*36^n+67)/5 (O{L}Z in base 36) currently at n=23729, no (probable) prime found. (6480*36^n+821)/7 ({P}SZ in base 36) currently at n=20235, no (probable) prime found. |
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[QUOTE=sweety439;531553](86*40^n+37)/3 (S{Q}d in base 40) currently at n=21939, no (probable) prime found.
(4428*36^n+67)/5 (O{L}Z in base 36) currently at n=23729, no (probable) prime found. (6480*36^n+821)/7 ({P}SZ in base 36) currently at n=20235, no (probable) prime found.[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) tested to n=25K, no (probable) prime found. Extended to n=50K |
(4428*36^n+67)/5 (O{L}Z in base 36) currently at n=32401, no (probable) prime found.
(6480*36^n+821)/7 ({P}SZ in base 36) currently at n=26743, no (probable) prime found. |
[QUOTE=sweety439;531552]A (probable) prime was found:
(13998*40^12381+29)/13 Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X This number is likely the second-largest "base 40 minimal prime"[/QUOTE] See the page [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] |
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[QUOTE=sweety439;531599](86*40^n+37)/3 (S{Q}d in base 40) tested to n=25K, no (probable) prime found.
Extended to n=50K[/QUOTE] (86*40^n+37)/3 (S{Q}d in base 40) seems to have a low weight, for 25K<=n<=50K, sieve to p=10^9, only 481 n remain. |
[QUOTE=sweety439;531600](4428*36^n+67)/5 (O{L}Z in base 36) currently at n=32401, no (probable) prime found.
(6480*36^n+821)/7 ({P}SZ in base 36) currently at n=26743, no (probable) prime found.[/QUOTE] I know that they can be reduced to (123*36^n+67)/5 and (5*36^n+821)/7, however, we let n be the number of the digits in "{}" (thus, the base 40 unsolved family should be (3440*40^n+37)/3 .... |
We assume the conjecture in post [URL="https://mersenneforum.org/showpost.php?p=529838&postcount=675"]https://mersenneforum.org/showpost.php?p=529838&postcount=675[/URL] is true (thus, all families in the files "unsolved xx" in [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]https://github.com/curtisbright/mepn-data/tree/master/data[/URL] and all families in the files "left xx" in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] have infinitely many primes)
Then the number of base n digits of the largest base n minimal prime is about 2^eulerphi(n) [CODE] n length of the largest minimal prime in base n 2 2 3 3 4 2 5 5 6 5 7 5 8 9 9 4 10 8 11 45 12 8 13 32021 (PRP) 14 86 15 107 16 3545 18 33 20 449 22 764 23 800874 (PRP) 24 100 30 1024 42 487 [/CODE] [CODE] n excepted length of the largest minimal prime in base n 2 2 3 4 4 4 5 16 6 4 7 64 8 16 9 64 10 16 11 1024 12 16 13 4096 14 64 15 256 16 256 17 65536 18 64 19 262144 20 256 21 4096 22 1024 23 4194304 24 256 25 1048576 26 4096 27 262144 28 4096 29 268435456 30 256 31 1073741824 32 65536 33 1048576 34 65536 35 16777216 36 4096 37 68719476736 38 262144 39 16777216 40 65536 41 1099511627776 42 4096 43 4398046511104 44 1048576 45 16777216 46 4194304 47 70368744177664 48 65536 49 4398046511104 50 1048576 51 4294967296 52 16777216 53 4503599627370496 54 262144 55 1099511627776 56 16777216 57 68719476736 58 268435456 59 288230376151711744 60 65536 61 1152921504606846976 62 1073741824 63 68719476736 64 4294967296 65 281474976710656 66 1048576 67 73786976294838206464 68 4294967296 69 17592186044416 70 16777216 71 1180591620717411303424 72 16777216 [/CODE] |
Also, assume the conjecture in post [URL="https://mersenneforum.org/showpost.php?p=529838&postcount=675"]https://mersenneforum.org/showpost.php?p=529838&postcount=675[/URL] is true:
[CODE] n length of largest minimal prime in base n 17 >1000000 19 >707000 21 >506700 25 >660000 (because of the EF{O} family, given by [URL="https://github.com/curtisbright/mepn-data/blob/master/data/sieve.25.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/sieve.25.txt[/URL]) 26 >486700 27 >368000 28 >543000 29 >242300 31 >=524288 (because of the {F}G family, given by [URL="https://oeis.org/A275530"]https://oeis.org/A275530[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]) 32 >=3435973837 (because of the G{0}1 family, given by [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]) 33 >10000 34 >10000 35 >10000 36 >32401 (the only two unsolved families are both reserved by me) 37 >=22023 (because of the prime FY{a[SUB]22021[/SUB]}, given by CRUS) 38 >=16777217 (because of the 1{0}1 family, see [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL] and [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]) 39 >10000 40 >25000 (the only one unsolved family is reserved by me) 41 >10000 43 >10000 44 >10000 45 >=18523 (because of the prime O{0[SUB]18521[/SUB]}1, given by CRUS, note that the prime AO{0[SUB]44790[/SUB]}1 is not a minimal prime in base 45, although AO{0}1 is in [URL="https://github.com/RaymondDevillers/primes/blob/master/left45"]https://github.com/RaymondDevillers/primes/blob/master/left45[/URL]) 46 >250000 (because of the d4{0}1 family, given by CRUS) 47 >10000 48 >250000 (because of the a{0}1 family, given by CRUS) 49 >=52700 (because of the prime SL{m[SUB]52698[/SUB]}, given by CRUS) 50 >=16777217 (because of the 1{0}1 family, see [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL] and [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]) [/CODE] |
[QUOTE=sweety439;531551]Base 36 has only two unsolved family:
(4428*36^n+67)/5 (6480*36^n+821)/7 Base 40 has only two unsolved family: (13998*40^n+29)/13 (86*40^n+37)/3[/QUOTE] The two unsolved family should be: (559920*40^n+29)/13 (3440*40^n+37)/3 and this (probable) prime should be: (559920*40^12380+29)/13 (13998*40^12381+29)/13 is the reduced form |
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No (probable) prime found for (86*40^n+37)/3 (S{Q}d in base 40) for n=25K-50K.
Text file attached. Extended to n=100K. |
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Base 36:
O{L}Z (4428*36^n+67)/5: tested to n=50K, no (probable) prime found {P}SZ (6480*36^n+821)/7: currently at n=41566, no (probable) prime found Base 40: S{Q}d (86*40^n+37)/3: currently at n=59777, no (probable) prime found |
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