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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 2022-06-08 15:22

1 Attachment(s)
The product of all known minimal primes (start with b+1) base b: (I have added the "[URL="https://en.wikipedia.org/wiki/Aliquot_sequence"]Aliquot sequence[/URL]" "[URL="https://en.wikipedia.org/wiki/Home_prime"]Home prime[/URL] base b (only for the same base b)" "Inverse home prime base b (only for the same base b)" sequences in factordb starting with these numbers, they will be interesting!!! Also, I have already added "all these primes (or PRPs)" "all these primes (or PRPs)-1" "all these primes (or PRPs)+1" to factordb, as they are used as [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality proving[/URL] [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality proving[/URL] [URL="https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm"]P-1 integer factorization algorithm[/URL] [URL="https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm"]P+1 integer factorization algorithm[/URL])

(b = 13, 16, 17 have unsolved families)

(b = 11, 13, 16, 17, 22, 30 have unproven PRPs)

[URL="http://factordb.com/index.php?id=3"]b=2[/URL]

[URL="http://factordb.com/index.php?id=455"]b=3[/URL]

[URL="http://factordb.com/index.php?id=205205"]b=4[/URL]

[URL="http://factordb.com/index.php?id=1100000002457822814"]b=5[/URL]

[URL="http://factordb.com/index.php?id=1100000002457821560"]b=6[/URL]

[URL="http://factordb.com/index.php?id=1100000002457825324"]b=7[/URL]

[URL="http://factordb.com/index.php?id=1100000002371473795"]b=8[/URL]

[URL="http://factordb.com/index.php?id=1100000003450366253"]b=9[/URL]

[URL="http://factordb.com/index.php?id=1100000002370859491"]b=10[/URL]

[URL="http://factordb.com/index.php?id=1100000003583737715"]b=11[/URL]

[URL="http://factordb.com/index.php?id=1100000002457818232"]b=12[/URL]

[URL="http://factordb.com/index.php?id=1100000003782980695"]b=13[/URL]

[URL="http://factordb.com/index.php?id=1100000003575953976"]b=14[/URL]

[URL="http://factordb.com/index.php?id=1100000003588261354"]b=15[/URL]

[URL="http://factordb.com/index.php?id=1100000003793382618"]b=16[/URL]

[URL="http://factordb.com/index.php?id=1100000003798135482"]b=17[/URL]

[URL="http://factordb.com/index.php?id=1100000003590551018"]b=18[/URL]

[URL="http://factordb.com/index.php?id=1100000003590550972"]b=20[/URL]

[URL="http://factordb.com/index.php?id=1100000003798118963"]b=22[/URL]

[URL="http://factordb.com/index.php?id=1100000003782956267"]b=24[/URL]

[URL="http://factordb.com/index.php?id=1100000003782953720"]b=30[/URL]

 sweety439 2022-06-09 14:18

New link for the minimal set of the primes > b in base b: [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]https://github.com/xayahrainie4793/quasi-mepn-data[/URL]

This site includes bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, all are completely solved (if we allow probable primes in place of proven primes) except base 13 and base 16, and both of these two bases have 2 unsolved families, bases 11, 18, 20, 22, 24, 30 needs primality proving of the probable primes (base 18 and base 20 are easier, since their largest probable primes only have 7872 and 8159 decimal digits, respectively)

The minimal set of the primes > b in base b must includes: (if such prime exists in base b)

* The smallest prime of the form (b^n-1)/(b-1) with n>=2 (i.e. the smallest generalized repunit prime in base b), see [URL="https://oeis.org/A084740"]https://oeis.org/A084740[/URL], [URL="https://oeis.org/A084738"]https://oeis.org/A084738[/URL], [URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL], [URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL], [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL], [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html[/URL], [URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/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]
* The smallest prime of the form b^n+1 with n>=1 (i.e. the smallest generalized Fermat prime in base b), see [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], [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/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL]
* The smallest prime of the form 2*b^n-1 with n>=1 (see [URL="https://oeis.org/A119591"]https://oeis.org/A119591[/URL], [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=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL], [URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL])
* The smallest prime of the form b^n+2 with n>=1 (see [URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL], [URL="https://oeis.org/A084713"]https://oeis.org/A084713[/URL])
* The smallest prime of the form b^n-2 with n>=2 (see [URL="https://oeis.org/A250200"]https://oeis.org/A250200[/URL], [URL="https://oeis.org/A255707"]https://oeis.org/A255707[/URL], [URL="https://oeis.org/A084714"]https://oeis.org/A084714[/URL], [URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL])
* The smallest prime of the form (b-1)*b^n+1 with n>=1 (see [URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL], [URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL], [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.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL])
* The smallest prime of the form (b-1)*b^n-1 with n>=1 (see [URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL], [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL], [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="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="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL], [URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL])
* The smallest prime of the form b^n+(b-1) with n>=1 (see [URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL], [URL="https://oeis.org/A076846"]https://oeis.org/A076846[/URL], [URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL])
* The smallest prime of the form b^n-(b-1) with n>=2 (see [URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL], [URL="https://oeis.org/A343589"]https://oeis.org/A343589[/URL], [URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL], [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])

 xilman 2022-06-11 13:26

Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums\$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums\$
[/code]

Should be easy enough to fix but not a good advertisement ...

 xilman 2022-06-11 14:34

[QUOTE=xilman;607581]Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums\$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums\$
[/code]

Should be easy enough to fix but not a good advertisement ...[/QUOTE]Making it compile was easy but tedious.

However, the code is riddled with dubious string assignments which elicit numerous warnings from gcc. When the resulting binary runs the output is corrrect but incomplete. For instance, the output for base 6 is 11, 15, 21, 25, 31, 35, 45, 51 and longer strings are omitted. For base 10, nothing longer than 991 appears.

Not impressed.

 sweety439 2022-06-11 14:38

[QUOTE=xilman;607581]Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums\$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums\$
[/code]

Should be easy enough to fix but not a good advertisement ...[/QUOTE]

Well, this code need to run with GMP.

also, I contacted others, and he made a C++ code (with GMP) to run, and he ran this code for bases 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, but for bases 17, 19, 21, ..., this code seems to be very slow, he has sent the data to me, see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]my GitHub page[/URL]

 xilman 2022-06-11 14:44

[QUOTE=sweety439;607587]Well, this code need to run with GMP.

also, I contacted others, and he made a C++ code (with GMP) to run, and he ran this code for bases 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, but for bases 17, 19, 21, ..., this code seems to be very slow, he has sent the data to me, see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]my GitHub page[/URL][/QUOTE]Of course it was linked with GMP!

I do know how to write and build GMP-enabled programs.

To be honest, given that your code doesn't even compile, I had no confidence that it would do anything useful after being hacked to fix the egregious errors listed in my earlier post.

Why on earth should anyone want to check some of your numbers for primality, given this dismal record?

 sweety439 2022-06-11 14:49

[QUOTE=xilman;607588]Of course it was linked with GMP!

I do know how to write and build GMP-enabled programs.

To be honest, given that your code doesn't even compile, I had no confidence that it would do anything useful after being hacked to fix the egregious errors listed in my earlier post.

Why on earth should anyone want to check some of your numbers for primality, given this dismal record?[/QUOTE]

This code is not created by me, this is I copied others' code for the original minimal prime problem (i.e. prime > base is not needed) and try to update the code for my new problem (i.e. only consider the primes > base), but there may be errors.

 paulunderwood 2022-06-11 14:55

[QUOTE=sweety439;607589]This code is not created by me, this is I copied others' code for the original minimal prime problem (i.e. prime > base is not needed) and try to update the code for my new problem (i.e. only consider the primes > base), but there may be errors.[/QUOTE]

Hmm.

[CODE]minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)[/CODE]

Looks like [C]temp[/C] is [C]mpz_t[/C] and neither [C]mpz_cmp[/C] nor [C]mpz_cmp_ui[/C] were used for [C]temp > base[/C].

 xilman 2022-06-11 15:09

[QUOTE=paulunderwood;607590]Hmm.

[CODE]minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)[/CODE]

Looks like [C]temp[/C] is [C]mpz_t[/C] and neither [C]mpz_cmp[/C] nor [C]mpz_cmp_ui[/C] were used for [C]temp > base[/C].[/QUOTE]Exactly. The first one I fixed.

[code]
if(mpz_probab_prime_p(temp, 25) > 0 && mpz_cmp_si(temp, base) > 0) /* PCL */
[/code]

Note mpz_cmp_si --- base is declared int, not unsigned.

 sweety439 2022-06-24 06:09

[QUOTE=sweety439;606692]This problem (the minimal prime (start with b+1) problem) is better than the CRUS problem because:

* [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] exclude some k's having primes (e.g. R14 k=1, 1*14^1-1 is prime, and R20 k=1, 1*20^1-1 is prime), even though they in fact have primes (the reason is that they only have this prime and have no other primes), as well as the extended [URL="https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub"]Sierpinski[/URL] and [URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]Riesel[/URL] problems, S8 k=27 has a prime (27*8^1+1)/gcd(27+1,8-1), and S16 k=4 has a prime (4*16^1+1)/gcd(4+1,16-1), and R4 k=1, R8 k=1, R16 k=1, R36 k=1, etc. all of them have primes, but still excluded in the problems (the reason is that they only have this prime and have no other primes)), in contrast, although the prime "111" is the only prime in the family {1} in base 8, it is still included in the minimal prime (start with b+1) problem in base b=8, since the prime "111" in base 8 is not only the prime for the family {1} in base 8 but also the prime for the family 1{0}11 and 11{0}1 in base 8, the same holds for the prime "11" in base 4, the prime "11" in base 4 is not only the prime for the family {1} in base 4 but also the prime for the family 1{0}1 in base 4, a more complex example is the prime "G7" in base 27 (= 439 in decimal), this prime is the only prime in the family {G}7 in base 27 (because of the difference-of-cubes algebraic factorization), but this prime is still included, since this prime is also the prime for the family G{7} in base 27, and the family G{7} in base 27 may contain infinitely many primes.[/QUOTE]

This problem (the minimal prime (start with b+1) problem) covers these problems:

* Find the smallest prime of the form (b^n-1)/(b-1) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form (b^n+1)/2 (for odd b) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 2*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 2*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+2 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-2 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 3*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 3*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+3 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-3 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 4*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 4*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+4 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-4 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form (b-1)*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form (b-1)*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+(b-1) with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-(b-1) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form ((b-2)*b^n+1)/(b-1) with n>=2 (or prove that such primes do not exist)

 sweety439 2022-06-24 06:19

Now these minimal primes (start with b+1) in base b have been proven primes: (only list the numbers > 10^1000)

[CODE]
b index of this minimal prime in base b base-b form of the minimal prime algebraic ((a*b^n+c)/d) form of the minimal prime primality certificate for the minimal prime
9 151 3(0^1158)11 3×9^1160+10 [URL="http://factordb.com/cert.php?id=1100000002376318423"]http://factordb.com/cert.php?id=1100000002376318423[/URL]
11 1067 55(7^1011) (607×11^1011−7)/10 [URL="http://factordb.com/cert.php?id=1100000002361376522"]http://factordb.com/cert.php?id=1100000002361376522[/URL]
13 3184 (9^968)B (3×13^969+5)/4 [URL="http://factordb.com/cert.php?id=1100000000258566244"]http://factordb.com/cert.php?id=1100000000258566244[/URL]
13 3185 1(0^1295)181 13^1298+274 [URL="http://factordb.com/cert.php?id=1100000002615445013"]http://factordb.com/cert.php?id=1100000002615445013[/URL]
13 3186 (9^1362)5 (3×13^1363−19)/4 [URL="http://factordb.com/cert.php?id=1100000002321017776"]http://factordb.com/cert.php?id=1100000002321017776[/URL]
13 3187 (7^1504)1 (7×13^1505−79)/12 [URL="http://factordb.com/cert.php?id=1100000002320890755"]http://factordb.com/cert.php?id=1100000002320890755[/URL]
13 3188 93(0^1551)1 120×13^1552+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
13 3189 72(0^2297)2 93×13^2298+2 [URL="http://factordb.com/cert.php?id=1100000002632396910"]http://factordb.com/cert.php?id=1100000002632396910[/URL]
13 3190 177(0^2703)17 267×13^2705+20 [URL="http://factordb.com/cert.php?id=1100000003590430825"]http://factordb.com/cert.php?id=1100000003590430825[/URL]
13 3191 39(0^6266)1 48×13^6267+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
13 3192 B(0^6540)BBA 11×13^6543+2012 [URL="http://factordb.com/cert.php?id=1100000002616382906"]http://factordb.com/cert.php?id=1100000002616382906[/URL]
13 3193 (C^10631)92 13^10633−50 [URL="http://factordb.com/cert.php?id=1100000003590493750"]http://factordb.com/cert.php?id=1100000003590493750[/URL]
14 650 4(D^19698) 5×14^19698−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
16 2337 D(9^1052) (68×16^1052−3)/5 [URL="http://factordb.com/cert.php?id=1100000002321036020"]http://factordb.com/cert.php?id=1100000002321036020[/URL]
16 2338 FA(F^1062)45 251×16^1064−187 [URL="http://factordb.com/cert.php?id=1100000003588387610"]http://factordb.com/cert.php?id=1100000003588387610[/URL]
16 2339 F(8^1517)F (233×16^1518+97)/15 [URL="http://factordb.com/cert.php?id=1100000000633744824"]http://factordb.com/cert.php?id=1100000000633744824[/URL]
16 2340 2(0^1713)321 2×16^1716+801 [URL="http://factordb.com/cert.php?id=1100000003588386735"]http://factordb.com/cert.php?id=1100000003588386735[/URL]
16 2341 300(F^1960)AF 769×16^1962−81 [URL="http://factordb.com/cert.php?id=1100000003588368750"]http://factordb.com/cert.php?id=1100000003588368750[/URL]
16 2342 9(0^3542)91 9×16^3544+145 [URL="http://factordb.com/cert.php?id=1100000000633424191"]http://factordb.com/cert.php?id=1100000000633424191[/URL]
16 2343 5B(C^3700)D (459×16^3701+1)/5 [URL="http://factordb.com/cert.php?id=1100000000993764322"]http://factordb.com/cert.php?id=1100000000993764322[/URL]
18 549 C(0^6268)C5 12×18^6270+221 [URL="http://factordb.com/cert.php?id=1100000003590442437"]http://factordb.com/cert.php?id=1100000003590442437[/URL]
20 3312 5(0^1163)AJ 5×20^1165+219 [URL="http://factordb.com/cert.php?id=1100000003590502412"]http://factordb.com/cert.php?id=1100000003590502412[/URL]
20 3313 C(D^2449) (241×20^2449−13)/19 [URL="http://factordb.com/cert.php?id=1100000002325393915"]http://factordb.com/cert.php?id=1100000002325393915[/URL]
20 3314 G(0^6269)D 16×20^6270+13 [URL="http://factordb.com/cert.php?id=1100000003590539457"]http://factordb.com/cert.php?id=1100000003590539457[/URL]
22 7998 K(0^760)EC1 20×22^763+7041 [URL="http://factordb.com/cert.php?id=1100000000632724415"]http://factordb.com/cert.php?id=1100000000632724415[/URL]
22 7999 J(0^767)IGGJ 19×22^771+199779 [URL="http://factordb.com/cert.php?id=1100000003591362567"]http://factordb.com/cert.php?id=1100000003591362567[/URL]
22 8000 (7^959)K7 (22^961+857)/3 [URL="http://factordb.com/cert.php?id=1100000003591361817"]http://factordb.com/cert.php?id=1100000003591361817[/URL]
22 8001 (L^2385)KE7 22^2388−653 [URL="http://factordb.com/cert.php?id=1100000003591360774"]http://factordb.com/cert.php?id=1100000003591360774[/URL]
22 8002 (7^3815)2L (22^3817−289)/3 [URL="http://factordb.com/cert.php?id=1100000003591359839"]http://factordb.com/cert.php?id=1100000003591359839[/URL]
24 3405 (N^2644)LLN 24^2647−1201 [URL="http://factordb.com/cert.php?id=1100000003593270089"]http://factordb.com/cert.php?id=1100000003593270089[/URL]
24 3406 (D^2698)LD (13×24^2700+4403)/23 [URL="http://factordb.com/cert.php?id=1100000003593269876"]http://factordb.com/cert.php?id=1100000003593269876[/URL]
24 3407 A(0^2951)8ID 10×24^2954+5053 [URL="http://factordb.com/cert.php?id=1100000003593269654"]http://factordb.com/cert.php?id=1100000003593269654[/URL]
24 3408 88(N^5951) 201×24^5951−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
24 3409 N00(N^8129)LN 13249×24^8131−49 [URL="http://factordb.com/cert.php?id=1100000003593391606"]http://factordb.com/cert.php?id=1100000003593391606[/URL]
30 2616 C(0^1022)1 12×30^1023+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
30 2617 (5^4882)J (5×30^4883+401)/29 [URL="http://factordb.com/cert.php?id=1100000002327649423"]http://factordb.com/cert.php?id=1100000002327649423[/URL]
30 2619 O(T^34205) 25×30^34205−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
[/CODE]

and I also used the Windows version of Primo to prove the smaller minimal primes (start with b+1), see [URL="http://factordb.com/certoverview.php?userid=1266"]http://factordb.com/certoverview.php?userid=1266[/URL]

These numbers are only probable primes: (they are all unproven probable primes for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30) (all of them are [URL="https://primes.utm.edu/glossary/xpage/StrongPRP.html"]strong probable primes[/URL] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, and [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]trial factored[/URL] to 10^11

[CODE]
b index of this minimal prime in base b (assuming the primality of all probable primes in base b) base-b form of the unproven probable prime algebraic ((a*b^n+c)/d) form of the unproven probable prime
11 1068 5(7^62668) (57×11^62668−7)/10
13 3194 C(5^23755)C (149×13^23756+79)/12
13 3195 8(0^32017)111 8×13^32020+183
16 2344 D0(B^17804) (3131×16^17804−11)/15
16 2345 D(B^32234) (206×16^32234−11)/15
22 8003 B(K^22001)5 (251×22^22002−335)/21
30 2618 I(0^24608)D 18×30^24609+13
[/CODE]

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