A Sierpinski/Riesel-like problem - Page 98

sweety439 · 2020-10-19, 15:47

At n=26544, found a new (probable) prime: (376*70^24952-1)/3

sweety439 · 2020-10-22, 14:48

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

sweety439 · 2020-10-22, 14:59

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

sweety439 · 2020-10-25, 19:43

Quote:

Originally Posted by sweety439

No other (probable) primes found for R70 k = 376, 496, 811 up to n=22813

R70 at n=37634, no new (probable) prime found

sweety439 · 2020-10-28, 00:07

R70 at n=41326, no other (probable) primes found

sweety439 · 2020-10-28, 00:09

R43 at n=18122, no (probable) primes found

Unfortunately, srsieve cannot sieve R43, since k and b are both odd.

sweety439 · 2020-10-29, 00:49

Quote:

Originally Posted by sweety439

R70 at n=41326, no other (probable) primes found

R70 at n=43008, one new (probable) prime found: (376*70^42427-1)/3

Unfortunately, this does not help for the R70 problem, since k=376 already has the prime (376*70^6484-1)/3, and the only remain k (k=811) still does not have any (probable) primes

sweety439 · 2020-10-29, 03:28

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)

sweety439 · 2020-10-31, 00:22

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,b-1) and k*b^n-1 is always divisible by gcd(k-1,b-1), the formulas are (k*b^n+1)/gcd(k+1,b-1) for Sierpinski and (k*b^n-1)/gcd(k-1,b-1) 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,b-1) (for Sierpinski) or (k*b^n-1)/gcd(k-1,b-1) (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

sweety439 · 2020-10-31, 00:24

reserving S36 k=1814 for n = 87.5K - 100K, currently at n=92334, no (probable) prime found

sweety439 · 2020-10-31, 00:54

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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2020-10-22, 14:48	#1069
sweety439 Nov 2016 2²×3×5×47 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

2020-10-22, 14:59	#1070
sweety439 Nov 2016 2²×3×5×47 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

2020-10-29, 03:28	#1075
sweety439 Nov 2016 2²×3×5×47 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)

2020-10-31, 00:22	#1076
sweety439 Nov 2016 2²×3×5×47 Posts	These problems generalized the Sierpinski problem and the Riesel problem to other bases (instead of only base 2), since for bases b>2, kb^n+1 is always divisible by gcd(k+1,b-1) and kb^n-1 is always divisible by gcd(k-1,b-1), the formulas are (kb^n+1)/gcd(k+1,b-1) for Sierpinski and (kb^n-1)/gcd(k-1,b-1) for Riesel, for a given base b>=2, we will find and proof the smallest k>=1 such that (kb^n+1)/gcd(k+1,b-1) (for Sierpinski) or (kb^n-1)/gcd(k-1,b-1) (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

2020-10-31, 00:54	#1078
sweety439 Nov 2016 2²·3·5·47 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