20210607, 15:32  #1 
Nov 2016
2×11×131 Posts 
Smallest possible prime of the form (31^n+1)/2
I want to solve the generalized Sierpinski conjectures in bases 2<=b<=128, but for base b=31, the CK is 239, and there are 10 kvalues remaining with no known (probable) primes: {1, 43, 51, 73, 77, 107, 117, 149, 181, 209}, for k=1, the formula is (1*31^n+1)/2, and if this formula produce prime, then n must be power of 2 (since if n has an odd factor m>1, then (31^n+1)/2 is divisible by (31^m+1)/2, thus cannot be prime), for the status for (31^n+1)/2: (see http://factordb.com/index.php?query=...%29%2B1%29%2F2)
Code:
n factors 2^0 2^4 2^1 13*37 2^2 409*1129 2^3 17*P11 2^4 1889*... 2^5 4801*... 2^6 257*641*... 2^7 P58*P133 2^8 P11*P11*P361 2^9 25601*... 2^10 114689*... 2^11 composite 2^12 composite 2^13 1196033*... 2^14 4882433*... 2^15 65537*... 2^16 composite 2^17 composite (there is no n<11559 such that (n^(2^17)+1)/2 is prime, see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt) 2^18 255666946049*... 2^19 1775270625281*... 2^20 unknown Also for other Sierpinski bases with GFN (b^(2^n)+1) or half GFN ((b^(2^n)+1)/2) remain, such as 15 (k=225), 18 (k=18), 22 (k=22), 37 (k=37), 38 (k=1), 40 (k=1600), 42 (k=42), 50 (k=1), 52 (k=52), 55 (k=1), 58 (k=58), 60 (k=60)? What are the true test limit for these GFNs? I know that for all even bases, this test limits must be at least 2^231, see http://www.primegrid.com/stats_genefer.php and http://www.primegrid.com/forum_thread.php?id=3980 
20210607, 20:34  #2  
Jun 2012
Boulder, CO
283 Posts 
Quote:
1. Download sllr64: http://jpenne.free.fr/index2.html 2. Run it: Code:
./sllr64 d t8 q"(31^131072+1)/2" Starting probable prime test of (31^131072+1)/2 Using allcomplex AVX512 FFT length 32K, a = 3 31^131072+1)/2 is not prime. RES64: A26F6DFC06756BFA. OLD64: 92BDE0A717A043E9 Time : 42.352 sec. 

20210608, 09:41  #3 
"Oliver"
Sep 2017
Porta Westfalica, DE
23^{2} Posts 
On FactorDB, there are even factors. For both numbers!

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