A Sierpinski/Riesel-like problem

[sweety439]

Update newest word files.

[sweety439]

Originally Posted by kar_bon

Ok, now the next steps. Insert the other bases and results.

I would prefer the alternate view: if conditions and/or comments are given, the table entries are broadens too much (see Riesel base 12).

Note to remaining lists (see Sierp base 6):
create subpages to "Sierpinski problem (extended definition)" like "Sierpinski problem (extended definition)/base 6" because the phrase "Extended Sierpinski" is used for another project.

Don't forget to categorize new pages (see "Riesel problem (extended definition)").

For OEIS-sequences in "Repunit" there's a template, see "Template:OEIS".

And here's a new result: Riesel base 7 (extended), k-val = 197
(197*7^181761-1)/2 is PRP, 153608 digits

pfgw -q"(197*7^181761-1)/2"
PFGW Version 3.3.6.20100908.Win_Stable [GWNUM 25.14]
(197*7^181761-1)/2 is 3-PRP!

pfgw -tp -q"(197*7^181761-1)/2"
PFGW Version 3.3.6.20100908.Win_Stable [GWNUM 25.14]
Primality testing (197*7^181761-1)/2 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(197*7^181761-1)/2 is Lucas PRP!

The reason that I edit the "repunit" article is that I want to find the smallest odd prime p such that (b^p-1)/(b-1) is prime for all bases 2<=b<=1024, and the smallest odd prime p such that (b^p+1)/(b+1) is prime for all bases 2<=b<=1024, see http://mersenneforum.org/showthread.php?t=21808. There are 56 bases 2<=b<=1024 without known prime of the form (b^p-1)/(b-1) with odd prime p, these bases are 184, 185, 200, 210, 269, 281, 306, 311, 326, 331, 371, 380, 384, 385, 394, 396, 452, 465, 485, 511, 522, 570, 574, 598, 601, 629, 631, 632, 636, 640, 649, 670, 684, 691, 693, 711, 713, 731, 752, 759, 771, 795, 820, 861, 866, 872, 881, 932, 938, 948, 951, 956, 963, 996, 1005, 1015.

[sweety439]

Originally Posted by sweety439

The reason that I edit the "repunit" article is that I want to find the smallest odd prime p such that (b^p-1)/(b-1) is prime for all bases 2<=b<=1024, and the smallest odd prime p such that (b^p+1)/(b+1) is prime for all bases 2<=b<=1024, see http://mersenneforum.org/showthread.php?t=21808. There are 56 bases 2<=b<=1024 without known prime of the form (b^p-1)/(b-1) with odd prime p, these bases are 184, 185, 200, 210, 269, 281, 306, 311, 326, 331, 371, 380, 384, 385, 394, 396, 452, 465, 485, 511, 522, 570, 574, 598, 601, 629, 631, 632, 636, 640, 649, 670, 684, 691, 693, 711, 713, 731, 752, 759, 771, 795, 820, 861, 866, 872, 881, 932, 938, 948, 951, 956, 963, 996, 1005, 1015.

The top 10 known (probable) primes are (only sorted by exponent):

(152^270217-1)/151, (18^25667-1)/17, (487^9967-1)/486, (333^9743-1)/332, (391^9623-1)/390, (541^8951-1)/540, (907^7331-1)/906, (536^6653-1)/535, (922^5987-1)/921, (469^5987-1)/468.

[sweety439]

Originally Posted by gd_barnes

I will do that when you show that you are serious by (1) creating a web page to show all of this info. -and- (2) posting links to the latest software to use for sieving and searching.

Regardless base 63 has a huge conjecture on both sides. The entire primes file would be multiple gigabytes. I will still consider sending base 63 if you will do the above.

I have created web pages to show all of this info, see http://www.mersennewiki.org/index.ph..._definition%29 and http://www.mersennewiki.org/index.ph..._definition%29.

[sweety439]

Originally Posted by gd_barnes

I will do that when you show that you are serious by (1) creating a web page to show all of this info. -and- (2) posting links to the latest software to use for sieving and searching.

Regardless base 63 has a huge conjecture on both sides. The entire primes file would be multiple gigabytes. I will still consider sending base 63 if you will do the above.

@Gary, please see the post #232.

[kar_bon]

Originally Posted by sweety439

Great!!! What about S10, k=269?

To give you a number of the work that stands for finding (197*7^181761-1)/2 as PRP:

all timings of those ~1500 checked candidates for PRP with pfgw on a i7-2600 3,4 GHz 64Bit stystem doing in one core took me ~158 hours and those candidates eliminated by trial factoring are not included here.

I got my own searches and factoring and this was only of some interest to find some high PRPs.

Your researches on your own RS-conj. are at n=1K or 6K only by now, so do some work say n=50K for all bases. Primes searching and especially those projects need patience and the results will come.

The problem on your project is (as Gary mentioned), you have to determine the GCD for every k-value of any base to search for. It would by better to give those GCDs in the tables: sorting k-values by GCD.

[sweety439]

Originally Posted by kar_bon

To give you a number of the work that stands for finding (197*7^181761-1)/2 as PRP:

all timings of those ~1500 checked candidates for PRP with pfgw on a i7-2600 3,4 GHz 64Bit stystem doing in one core took me ~158 hours and those candidates eliminated by trial factoring are not included here.

I got my own searches and factoring and this was only of some interest to find some high PRPs.

Your researches on your own RS-conj. are at n=1K or 6K only by now, so do some work say n=50K for all bases. Primes searching and especially those projects need patience and the results will come.

The problem on your project is (as Gary mentioned), you have to determine the GCD for every k-value of any base to search for. It would by better to give those GCDs in the tables: sorting k-values by GCD.

Thus, will you reserve S10, k=269?

You can report the two PRPs you found ((197*7^181761-1)/2 and (1654*30^38869-1)/29) at http://www.primenumbers.net/prptop/prptop.php.

This GCD is very easy to compute, it is just gcd(k+1,b-1) (for extended Sierpinski problems) or gcd(k-1,b-1) (for extended Riesel problems).

[sweety439]

Update the text file for the conjectured k's for all bases 2<=b<=64, now, I am looking for the conjectured k's for bases b>64.

[sweety439]

This is the text file for the conjectured k's for all bases 2<=b<=128. (the conjectured k's for some bases are unknown, all of them are > 20000, these bases are S66, S78, S96, S108, S120, S124, S126, S127, R66, R78, R82, R96, R106, R120, R124, R126, R127)

[sweety439]

Originally Posted by sweety439

These problems are finding and proving the smallest k such that (k*b^n+-1)/gcd(k+-1, b-1) (+ for Sierpinski, - for Riesel) is composite for all natural numbers n>=1, for a given natural number base b>=2.

gcd(k+-1, b-1) is the largest number that divides k*b^n+-1 for all n.

Note: gcd(0, m) = m for all positive integer m, and gcd(1, m) = 1 for all integer m.

[sweety439]

Of course, we can also find the primes for the k's > CK. e.g. we can try to prove the 2nd conjecture, 3rd conjecture, 4th conjecture, ..., for a fixed Sierpinski/Riesel base.

For example, the 2nd conjectured k for S2 is 271129, and this conjecture is being worked on https://www.primegrid.com/forum_thread.php?id=1750. Besides, the 2nd conjectured k for R4 is 919, and this conjecture is proven, but with one non-certified probable prime (751*4^6615-1)/3. (for the (probable) prime for the 2nd conjecture for R4, see post #81)

However, in this project, we only decide to prove the "1st conjecture". Thus, in this project, we only consider the k's < CK.