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-   -   Smallest possible prime of the form (31^n+1)/2 (https://www.mersenneforum.org/showthread.php?t=26879)

 sweety439 2021-06-07 15:32

Smallest possible prime of the form (31^n+1)/2

I want to solve the [URL="https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub"]generalized Sierpinski conjectures in bases 2<=b<=128[/URL], but for base b=31, the CK is 239, and there are 10 k-values 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 [URL="http://factordb.com/index.php?query=%2831%5E%282%5En%29%2B1%29%2F2"]http://factordb.com/index.php?query=%2831%5E%282%5En%29%2B1%29%2F2[/URL])

[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 [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])
2^18 255666946049*...
2^19 1775270625281*...
2^20 unknown
[/CODE]

Thus what is the true test limit for S31 k=1, is it 2^20-1 = 1048575? Can someone check whether (31^(2^20)+1)/2, (31^(2^21)+1)/2, etc. is probable prime or not?

Also for other [URL="https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub"]Sierpinski bases[/URL] 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 2^23-1, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL] and [URL="http://www.primegrid.com/forum_thread.php?id=3980"]http://www.primegrid.com/forum_thread.php?id=3980[/URL]

 ryanp 2021-06-07 20:34

[QUOTE=sweety439;580241]Can someone check whether (31^(2^20)+1)/2, (31^(2^21)+1)/2, etc. is probable prime or not?[/QUOTE]

Why can't you check them yourself?

2. Run it:

[CODE] ./sllr64 -d -t8 -q"(31^131072+1)/2"
Starting probable prime test of (31^131072+1)/2
Using all-complex AVX-512 FFT length 32K, a = 3

31^131072+1)/2 is not prime. RES64: A26F6DFC06756BFA.
OLD64: 92BDE0A717A043E9 Time : 42.352 sec.[/CODE]

 kruoli 2021-06-08 09:41

On FactorDB, there are even factors. For [I]both[/I] numbers!

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