20120912, 23:44  #1 
Aug 2012
19_{16} Posts 
Primes for proven bases
I think it would be interesting to look at distributions of primes of the form k*b^n (+/) 1. Perhaps some neat results could come out of it. At the least, neat pictures.
The data storage could be minimal if we put aside the gigantic ck numbers. We've already got tables of b and ck. Next, put together two other sets of tables (one set for Sierp, one set for Riesel), one for each b, where the columns are: k, n, doubleCheckedPrime Where n is the minimum such that k*b^n (+/) 1 is prime. And the additional flag is for making sure the primes are actually prime (not too difficult). Doublechecking that it's the min n would be a bit more work. Or stack the tables and throw in a column for b. If we start with only the proved conjectures, this entire table would only be like 50,000 rows, and I'd be glad to store/host that. We're doing a ton of work and getting cool results, but I personally would love to be able to play with the data and see what else pops out. After a few minutes of looking at the b vs. ck tables, for all b up to 1030, it looks like for b = 14 + 15*j, j a natural number, the ck is 4 on both the Sierp and Riesel side. Not sure if anyone noticed this before, or if it's been conjectured/proved in general, but I haven't noticed it through perusing the forum (though I haven't read through all the reservation thread posts). Maybe looking at the distribution of n for k = 2 and k = 3 for those types of bases will provide some cool insights. Maybe not. There are some other interesting periodic behaviors with other cks, too. On the proven conjectures page, there's the "largest prime," so I imagine a lot of this information already exists, I'm just not sure where to start consolidating from. 
20120913, 01:54  #2 
"Gary"
May 2007
Overland Park, KS
5·17·139 Posts 
I have attempted to store every prime ever found plus all results (residuals) for n>25K on this project on my laptop. It gets backed up about every 23 months on an external hard drive. There are holes of primes that I do not have for some huge conjectured bases like base 3 as well as some others where the primes were posted in the threads here but not copied off to my machine. All of these primes, results, and sieve files are in the 2025 GB range right now and we haven't even scratched the surface of primes to be found on monsterous conjectured bases like 3, 7, 15, and 280.
The main thing that is needed for historical reference is primes for all k for proven bases. With the largest conjectured k proven at k<2500, it is a very small amount of data to have all of such primes available. If you would like a list of all primes for any proven base or one in which there are only 1, 2, or 3 k's remaining, I should be able to quickly provide it. I agree that a doublecheck effort is probably needed at some point but where and how to start such an effort is not something that I care to think about right now. The 2 main challenges from my perspective on this project have been managing the huge amount of information that comes in and attempting to give the project a direction. Fortunately coadmins and some of our regulars have helped. Mark (rogue) started the 1k base thread and did extensive updates to our stats pages as well as started a mini drive for base S63. Myself and others have helped start and maintain team drives and done large amounts of sieving to keep the drives going. Gary Last fiddled with by gd_barnes on 20120913 at 01:55 
20120914, 08:13  #3 
"Gary"
May 2007
Overland Park, KS
5·17·139 Posts 
For historical reference, attached is a list of all primes for proven bases <= 200.
Last fiddled with by gd_barnes on 20120920 at 06:23 Reason: bases to 200 
20120920, 06:23  #4 
"Gary"
May 2007
Overland Park, KS
27047_{8} Posts 
I've now added primes for proven bases <= 200 to the attachment in the previous post.

20161124, 16:35  #5 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
The CRUS page only shows the "TOP 10" primes, and I cannot find any information for the smallest n such that 286*19^n+1, 12*282^n+1, 12*457^n+1, etc. is prime, I only know that there is a known prime of this form (since the CRUS page says that they are not remaining), and I cannot see the primes, I hope the CRUS page has a link to the files like them: (including all k<conjectured k, not only k<=500, not only for proven bases, it is for all tested bases, if this k is remaining, the file can list [200K], etc. (only lists the k's such that gcd(k+1, b1) = 1 (+ for Sierpinski,  for Riesel))
Last fiddled with by sweety439 on 20161124 at 16:42 
20161125, 04:32  #6 
Romulan Interpreter
"name field"
Jun 2011
Thailand
19×541 Posts 
Something is "odd" there, in R2 and S2 you should not have "even" k's, and also in the other, try keep also gcd(k,b)=1. Any reason why they are kept?

20161125, 06:01  #7 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
gcd(k,b) does not need to be 1, only gcd(k+1, b1) needs to be 1. If gcd(k+1, b1) != 1, then this k is a trivial k (all numbers are divisible by a single prime factor)
Besides, when gcd(k,b) != 1, k may not be a multiple of b, for example 7666*10^n+1, gcd(7666,10) = 2, but 7666 is not a multiple of 10. Thus, 7666*10^n+1 needs to be tested. Even if k is a multiple of b, some k's still need to be tested: (when k+1 is prime, + for Sierpinski,  for Riesel) e.g. 6977*2^3+1 is prime, but it is hard to find an n>=1 such that 55816*2^n+1 is prime. e.g. 337*2^11 is prime, but it is hard to find an n>=1 such that 674*2^n1 is prime. e.g. 8*5^1+1 is prime, but it is hard to find an n>=1 such that 40*5^n+1 is prime. e.g. 36*6^2+1 is prime, but it is hard to find an n>=1 such that 1296*6^n+1 is prime. e.g. 45*10^11 is prime, but it is hard to find an n>=1 such that 450*10^n1 is prime. e.g. 22*27^11 is prime, but it is hard to find an n>=1 such that 594*27^n1 is prime. If k is a multiple of b and k+1 (+ for Sierpinski,  for Riesel) is not prime, then the prime for the k is the same as the prime for k/b. Thus, such k's do not need to be tested. Last fiddled with by sweety439 on 20161125 at 06:32 
20161125, 06:20  #8 
Romulan Interpreter
"name field"
Jun 2011
Thailand
19·541 Posts 
Ok, understood. A bit bad wording on my side, I wanted to say that you do not need to keep in the list the k's which contain multiplies of b. But I understood the reason meantime, by reading your discussion with Gary in the other thread.

20161125, 06:23  #9  
"Gary"
May 2007
Overland Park, KS
5·17·139 Posts 
Quote:


20161129, 23:44  #10  
"Gary"
May 2007
Overland Park, KS
10111000100111_{2} Posts 
Quote:
286*19^18524+1 12*282^2956+1 12*457^10023+1 

20161130, 13:28  #11 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
Thanks.
Yesterday, you gave all the Sierpinski primes in the CRUS with k=10. Also, in the other files, you gave all the Sierpinski/Riesel prime (including some not in the CRUS, i.e. the bases not started or the k > conjectured k) for all 2<=k<=7. I also want all 8<=k<=12 (only primes in the CRUS), especially Sierpinski primes with k=12, OK? Besides, In my research for reverse Sierpinski/Riesel problem for 2<=k<=24, I need these n's of these form, these are the status of this problem: (only for 8<=k<=24, since you had already given the status of 2<=k<=7) 9*244^n+1 (the conjecture of S244 is only 6) 9*248^39510+1 due to the CRUS 9*490^n+1 (S490 has 29 k's remaining at n=100K, but not include 9) 9*544^4705+1 due to the CRUS 9*592^n+1 (S592 has k=9 remaining at n=25K) 9*724^n+1 (S724 has k=9 remaining at n=200K) 9*844^9687+1 due to the CRUS 9*848^n+1 (S848 has 11 k's remaining at n=100K, but not include 9) 9*884^n+1 (the conjecture of S884 is only 4) 9*894^3069+1 due to the CRUS 9*908^n+1 (S908 has 9 k's remaining at n=100K, but not include 9) 9*934^429+1 due to the CRUS 9*974^n+1 (the conjecture of S974 is only 4) 9*984^n+1 (S984 has 6 k's remaining at n=100K, but not include 9) 9*1004^n+1 (the conjecture of S1004 is only 4) 9*1030^n+1 (S1030 has 304 k's remaining at n=25K, but not include 9) 10*17^1356+1 due to the CRUS 12*12^n+1 (it is a GFN) 12*30^1023+1 due to the CRUS 12*65^684+1 (this prime is found by me) 12*68^656921+1 due to the CRUS 12*87^1214+1 due to the CRUS 12*102^2739+1 due to the CRUS 15*398^n+1 (the conjecture of S398 is only 8) 15*496^44172+1 due to the CRUS 15*636^9850+1 due to the CRUS 15*650^n+1 (the conjecture of S650 is only 8) 15*734^n+1 (the conjecture of S734 is only 4) 15*752^1128+1 due to the CRUS 15*846^408+1 due to the CRUS 15*864^51510+1 due to the CRUS 15*874^n+1 (the conjecture of S874 is only 6) 15*876^n+1 (S876 is a nonstarted base) 15*1014^n+1 (the conjecture of S1014 is only 6) 16*32^n+1 (it is a GFN) 17*68^n+1 (S68 has k=17 remaining at n=1M) 17*218^n+1 (S218 has k=17 remaining at n=200K) 18*18^n+1 (it is a GFN) 18*37^461+1 (this prime is found by me) 18*74^n+1 (the conjecture of S74 is only 4) 18*145^n+1 (S145 has 435 k's remaining at n=25K, but not include 18) 18*157^3873+1 due to the CRUS 18*189^171175+1 due to the CRUS 18*227^n+1 (S227 has k=18 remaining at n=1M) 18*239^n+1 (the conjecture of S239 is only 4) 18*293^n+1 (the conjecture of S293 is only 8) 22*22^n+1 (it is a GFN) 24*45^n+1 (S45 has 36 k's remaining at n=100K, but not include 24) 24*79^n+1 (S79 has k=24 remaining at n=200K) 12*65^n1 (the conjecture of R65 is only 10) 12*98^n1 (the conjecture of R98 is only 10) 15*454^n1 (the conjecture of R454 is only 6) 15*466^n1 (R466 has 54 k's remaining at n=100K, but not include 15) 15*552^n1 (R552 has k=15 remaining at n=100K) 15*608^n1 (the conjecture of R608 is only 8) 15*620^5621 due to the CRUS 15*718^n1 (R718 has 22 k's remaining at n=100K, but not include 15) 15*734^n1 (the conjecture of R734 is only 4) 15*774^19371 due to the CRUS 15*828^23081 due to the CRUS 15*856^n1 (R856 is a nonstarted base) 17*88^n1 (R88 has 30 k's remaining at n=100K, but not include 17) 17*98^n1 (the conjecture of R98 is only 10) 17*110^25981 due to the CRUS 17*556^n1 (R556 is a nonstarted base) 17*650^n1 (the conjecture of R650 is only 8) 17*662^n1 (the conjecture of R662 is only 14) 17*686^4021 due to the CRUS 17*724^10821 due to the CRUS 17*734^n1 (the conjecture of R734 is only 4) 17*766^n1 (R766 has 6 k's remaining at n=100K, but not include 17) 17*772^n1 (R772 has 587 k's remaining at n=25K, but not include 17) 17*842^356401 due to the CRUS 17*852^n1 (R852 has 130 k's remaining at n=25Km, but not include 17) 17*988^12751 due to the CRUS 22*38^n1 (the conjecture of R38 is only 13) 22*62^n1 (the conjecture of R62 is only 8) 24*45^1533551 due to the CRUS 24*64^30201 (this prime is found by me) 24*72^26481 due to the CRUS Last fiddled with by sweety439 on 20161130 at 14:10 
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