20201019, 15:47  #1068 
Nov 2016
2^{2}×3×5×47 Posts 
At n=26544, found a new (probable) prime: (376*70^249521)/3

20201022, 14:48  #1069 
Nov 2016
2^{2}×3×5×47 Posts 
For the Sierpinski bases 2<=b<=128 and b = 256, 512, 1024:
proven: 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 20, 21, 23, 27, 29, 34, 35, 39, 41, 43, 44, 45, 47, 49, 51, 54, 56, 57, 59, 61, 64, 65, 69, 71, 73 (with PRP), 74, 75, 76, 79, 84, 85, 87, 88, 90, 94, 95, 100, 101, 105 (with PRP), 110, 111, 114, 116, 119, 121, 125, 256 (with PRP) weak proven (only GFN's or half GFN's remain): 12, 18, 32, 37, 38, 50, 55, 62, 72, 77, 89, 91, 92, 98, 99, 104, 107, 109 1k bases: 25, 53, 103, 113, 118 2k bases but become 1k bases if GFN's and half GFN's are excluded: 10, 36, 68, 83, 86, 117, 122, 128 4k bases but become 1k bases if GFN's and half GFN's are excluded: 512 2k bases: 26, 30, 33, 46, 115 3k bases but become 2k bases if GFN's and half GFN's are excluded: 67, 123 3k bases: 28, 102 4k bases but become 3k bases if GFN's and half GFN's are excluded: 93 5k bases but become 3k bases if GFN's and half GFN's are excluded: 1024 
20201022, 14:59  #1070 
Nov 2016
2^{2}×3×5×47 Posts 
For the Riesel bases 2<=b<=128 and b = 256, 512, 1024:
proven: 4, 5, 7 (with PRP), 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 25, 26, 27, 29, 32, 34, 35, 37, 38, 39, 41, 44, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 59, 62, 64, 65, 67, 68, 69, 71, 72, 73 (with PRP), 74, 75, 76, 77, 79, 81, 83, 84, 86, 87, 89, 90, 91 (with PRP), 92, 95, 98, 99, 100 (with PRP), 101, 103, 104, 107 (with PRP), 109, 110, 111, 113, 114, 116, 119, 121, 122, 125, 128, 256, 512 1k bases: 43, 70, 85, 94, 97, 118, 123 2k bases: 33, 105, 115 3k bases: 46, 61, 80 
20201025, 19:43  #1071 
Nov 2016
2^{2}×3×5×47 Posts 

20201028, 00:07  #1072 
Nov 2016
2^{2}×3×5×47 Posts 
R70 at n=41326, no other (probable) primes found

20201028, 00:09  #1073 
Nov 2016
2820_{10} Posts 
R43 at n=18122, no (probable) primes found
Unfortunately, srsieve cannot sieve R43, since k and b are both odd. 
20201029, 00:49  #1074 
Nov 2016
2820_{10} Posts 
R70 at n=43008, one new (probable) prime found: (376*70^424271)/3
Unfortunately, this does not help for the R70 problem, since k=376 already has the prime (376*70^64841)/3, and the only remain k (k=811) still does not have any (probable) primes Last fiddled with by sweety439 on 20201029 at 00:50 
20201029, 03:28  #1075 
Nov 2016
2^{2}×3×5×47 Posts 
If k is rational power of base (b), then .... (let k = b^(r/s) with gcd(r,s) = 1)
* For the Riesel case, this is generalized repunit number to base b^(1/s) * For the Sierpinski case, if s is odd, then this is generalized (half) Fermat number to base b^(1/s) * For the Sierpinski case, if s is even, then this is generalized repunit number to negative base b^(1/s) 
20201031, 00:22  #1076 
Nov 2016
2^{2}×3×5×47 Posts 
These problems generalized the Sierpinski problem and the Riesel problem to other bases (instead of only base 2), since for bases b>2, k*b^n+1 is always divisible by gcd(k+1,b1) and k*b^n1 is always divisible by gcd(k1,b1), the formulas are (k*b^n+1)/gcd(k+1,b1) for Sierpinski and (k*b^n1)/gcd(k1,b1) for Riesel, for a given base b>=2, we will find and proof the smallest k>=1 such that (k*b^n+1)/gcd(k+1,b1) (for Sierpinski) or (k*b^n1)/gcd(k1,b1) (for Riesel) is not prime for all n>=1, any k that obtains a full covering set in any manner from ALGEBRAIC factors will be excluded, in many instances, this includes k's where there is a partial covering set of numeric factors (or a single numeric factor) and a partial covering set of algebraic factors that combine to make a full covering set, all k below the lowest k found to have a NUMERIC covering set must have a prime including multiples of the base (MOB) but excluding the k's that obtains a full covering set in any manner from ALGEBRAIC factors, for the lowest k found to have a NUMERIC covering set for all bases b<=2048 and b = 4096, 8192, 16384, 32768, 65536, see Sierpinski and Riesel

20201031, 00:24  #1077 
Nov 2016
2^{2}×3×5×47 Posts 
reserving S36 k=1814 for n = 87.5K  100K, currently at n=92334, no (probable) prime found

20201031, 00:54  #1078 
Nov 2016
2^{2}·3·5·47 Posts 
For Riesel problem base b, k=1 proven composite by algebra factors if and only if b is perfect power (of the form m^r with r>1)
For Sierpinski problem base b, k=1 proven composite by algebra factors if and only if b is perfect odd power (of the form m^r with odd r>1) In Riesel problem base b, k=1 can only have prime for n which is prime In Sierpinski problem base b, k=1 can only have prime for n which is power of 2 
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