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

[sweety439]

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 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 http://factordb.com/index.php?query=...%29%2B1%29%2F2)

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

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 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 2^23-1, see http://www.primegrid.com/stats_genefer.php and http://www.primegrid.com/forum_thread.php?id=3980

[ryanp]

Originally Posted by sweety439

Can someone check whether (31^(2^20)+1)/2, (31^(2^21)+1)/2, etc. is probable prime or not?

Why can't you check them yourself?

1. Download sllr64: http://jpenne.free.fr/index2.html
2. Run it:

 ./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.

[kruoli]

On FactorDB, there are even factors. For both numbers!

[sweety439]

I verified that (31^(2^n)+1)/2 is composite for n<=23, see http://factordb.com/index.php?query=...at=1&sent=Show, (31^(2^22)+1)/2 has factor 238*2^22+1, and (31^(2^23)+1)/2 has factor 252860*2^23+1, but (31^(2^24)+1)/2 has no prime factor < 2^48, also (31^(2^25)+1)/2 has factor 7180016*2^25+1

[sweety439]

Originally Posted by ryanp

Why can't you check them yourself?

1. Download sllr64: http://jpenne.free.fr/index2.html
2. Run it:

 ./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.

I download this and run:

cllr.exe -q"(57*11^62668-7)/10"

But why it cannot work?

(For PFGW, I already know how to run)