mersenneforum.org A Sierpinski/Riesel-like problem
 Register FAQ Search Today's Posts Mark Forums Read

2020-06-14, 10:51   #815
sweety439

Nov 2016

5·571 Posts

Quote:
 Originally Posted by sweety439

Corrected: R96 has some k proven composite by partial algebra factors

2020-06-17, 17:25   #816
sweety439

Nov 2016

1011001001112 Posts

the conjectured first 4 Sierpinski/Riesel numbers for bases up to 256
Attached Files
 conjectured first 4 Sierpinski numbers.txt (6.1 KB, 47 views) conjectured first 4 Riesel numbers.txt (6.1 KB, 43 views)

2020-06-18, 19:37   #817
sweety439

Nov 2016

5×571 Posts

Quote:
 Originally Posted by carpetpool (k*b^n+c)/gcd(k+c, b-1) is not a polynomial sequence so it isn't at all related to the Bunyakovsky conjecture. Steps 1 and 2 are trivial enough, but steps 3 and 4 are what make the difference. Step 4 isn't even part of the Bunyakovsky conjecture since that would imply a polynomial f(x) is irreducible over the integers. However, it can be proven for exponential sequences. For polynomial sequences, it's easy to prove Step 3: f(0) = C is the constant of a polynomial, so the infinitude of primes is implied by finding an integer x with gcd(x,C)=1 and gcd(f(x),C)=1. As for exponential type sequences, we can only assume that there "appear" to be infinitely many primes, and we can't prove if there exists a "covering set" or not. For example, we can't prove there doesn't exist a covering set for the sequence "3*2^n+-1", although it is extremely unlikely it exists. An exception, however, is divisibility sequences. For example, 2^n-1 does not have a covering set --- and we can prove this by showing that gcd(2^n-1,f)=1 for any prime f
Another conjecture:

If there are at least two primes of the form (k*b^n+c)/gcd(k+c, b-1) (k>=1, b>=2, c != 0, gcd(k,c)=1, gcd(b,c)=1) with n>=1, then (k*b^n+c)/gcd(k+c, b-1) has no covering set.

Strong conjecture:

If there are at least two primes of the form (k*b^n+c)/gcd(k+c, b-1) (k>=1, b>=2, c != 0, gcd(k,c)=1, gcd(b,c)=1) with n>=1, then (k*b^n+c)/gcd(k+c, b-1) satisfies step 3 (i.e. does not make a full covering set with (all primes), (all algebraic factors), or (partial primes, partial algebraic factors). (note that this is not true when there is only one such prime, counterexamples: (1*4^n-1)/gcd(1-1,4-1), (1*8^n-1)/gcd(1-1,8-1), (1*16^n-1)/gcd(1-1,16-1), (1*36^n-1)/gcd(1-1,36-1), (27*8^n+1)/gcd(27+1,8-1), ...)

If the strong conjecture and the conjecture in post #783 are both true, then:

If there are at least two primes of the form (k*b^n+c)/gcd(k+c, b-1) (k>=1, b>=2, c != 0, gcd(k,c)=1, gcd(b,c)=1) with n>=1, then there are infinitely many primes of this form.

 2020-06-19, 09:06 #818 sweety439     Nov 2016 5×571 Posts Stronger conjectures: (assuming k is positive integer) * if k < 78557, then there are infinitely many primes of the form (k*2^n+1)/gcd(k+1,2-1) with integer n>=1 * if k < 11047, then there are infinitely many primes of the form (k*3^n+1)/gcd(k+1,3-1) with integer n>=1 * if k < 419, then there are infinitely many primes of the form (k*4^n+1)/gcd(k+1,4-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*5^n+1)/gcd(k+1,5-1) with integer n>=1 * if k < 174308, then there are infinitely many primes of the form (k*6^n+1)/gcd(k+1,6-1) with integer n>=1 * if k < 209, then there are infinitely many primes of the form (k*7^n+1)/gcd(k+1,7-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*8^n+1)/gcd(k+1,8-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*9^n+1)/gcd(k+1,9-1) with integer n>=1 * if k < 989, then there are infinitely many primes of the form (k*10^n+1)/gcd(k+1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n+1)/gcd(k+1,11-1) with integer n>=1 * if k < 521, then there are infinitely many primes of the form (k*12^n+1)/gcd(k+1,12-1) with integer n>=1 * if k < 15, then there are infinitely many primes of the form (k*13^n+1)/gcd(k+1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n+1)/gcd(k+1,14-1) with integer n>=1 * if k < 673029, then there are infinitely many primes of the form (k*15^n+1)/gcd(k+1,15-1) with integer n>=1 * if k < 38, then there are infinitely many primes of the form (k*16^n+1)/gcd(k+1,16-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*17^n+1)/gcd(k+1,17-1) with integer n>=1 * if k < 398, then there are infinitely many primes of the form (k*18^n+1)/gcd(k+1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n+1)/gcd(k+1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n+1)/gcd(k+1,20-1) with integer n>=1 * if k < 23, then there are infinitely many primes of the form (k*21^n+1)/gcd(k+1,21-1) with integer n>=1 * if k < 2253, then there are infinitely many primes of the form (k*22^n+1)/gcd(k+1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n+1)/gcd(k+1,23-1) with integer n>=1 * if k < 30651, then there are infinitely many primes of the form (k*24^n+1)/gcd(k+1,24-1) with integer n>=1 * if k < 79, then there are infinitely many primes of the form (k*25^n+1)/gcd(k+1,25-1) with integer n>=1 * if k < 221, then there are infinitely many primes of the form (k*26^n+1)/gcd(k+1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n+1)/gcd(k+1,27-1) with integer n>=1 * if k < 4554, then there are infinitely many primes of the form (k*28^n+1)/gcd(k+1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n+1)/gcd(k+1,29-1) with integer n>=1 * if k < 867, then there are infinitely many primes of the form (k*30^n+1)/gcd(k+1,30-1) with integer n>=1 * if k < 239, then there are infinitely many primes of the form (k*31^n+1)/gcd(k+1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n+1)/gcd(k+1,32-1) with integer n>=1 * if k < 511, then there are infinitely many primes of the form (k*33^n+1)/gcd(k+1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n+1)/gcd(k+1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n+1)/gcd(k+1,35-1) with integer n>=1 * if k < 1886, then there are infinitely many primes of the form (k*36^n+1)/gcd(k+1,36-1) with integer n>=1 * if k < 39, then there are infinitely many primes of the form (k*37^n+1)/gcd(k+1,37-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*38^n+1)/gcd(k+1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n+1)/gcd(k+1,39-1) with integer n>=1 * if k < 47723, then there are infinitely many primes of the form (k*40^n+1)/gcd(k+1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n+1)/gcd(k+1,41-1) with integer n>=1 * if k < 13372, then there are infinitely many primes of the form (k*42^n+1)/gcd(k+1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n+1)/gcd(k+1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n+1)/gcd(k+1,44-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*45^n+1)/gcd(k+1,45-1) with integer n>=1 * if k < 881, then there are infinitely many primes of the form (k*46^n+1)/gcd(k+1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n+1)/gcd(k+1,47-1) with integer n>=1 * if k < 1219, then there are infinitely many primes of the form (k*48^n+1)/gcd(k+1,48-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*49^n+1)/gcd(k+1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n+1)/gcd(k+1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n+1)/gcd(k+1,51-1) with integer n>=1 * if k < 28674, then there are infinitely many primes of the form (k*52^n+1)/gcd(k+1,52-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*53^n+1)/gcd(k+1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n+1)/gcd(k+1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n+1)/gcd(k+1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n+1)/gcd(k+1,56-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*57^n+1)/gcd(k+1,57-1) with integer n>=1 * if k < 488, then there are infinitely many primes of the form (k*58^n+1)/gcd(k+1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n+1)/gcd(k+1,59-1) with integer n>=1 * if k < 16957, then there are infinitely many primes of the form (k*60^n+1)/gcd(k+1,60-1) with integer n>=1 * if k < 63, then there are infinitely many primes of the form (k*61^n+1)/gcd(k+1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n+1)/gcd(k+1,62-1) with integer n>=1 * if k < 1589, then there are infinitely many primes of the form (k*63^n+1)/gcd(k+1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n+1)/gcd(k+1,64-1) with integer n>=1 * if k < 509203, then there are infinitely many primes of the form (k*2^n-1)/gcd(k-1,2-1) with integer n>=1 * if k < 12119, then there are infinitely many primes of the form (k*3^n-1)/gcd(k-1,3-1) with integer n>=1 * if k < 361, then there are infinitely many primes of the form (k*4^n-1)/gcd(k-1,4-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*5^n-1)/gcd(k-1,5-1) with integer n>=1 * if k < 84687, then there are infinitely many primes of the form (k*6^n-1)/gcd(k-1,6-1) with integer n>=1 * if k < 457, then there are infinitely many primes of the form (k*7^n-1)/gcd(k-1,7-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*8^n-1)/gcd(k-1,8-1) with integer n>=1 * if k < 41, then there are infinitely many primes of the form (k*9^n-1)/gcd(k-1,9-1) with integer n>=1 * if k < 334, then there are infinitely many primes of the form (k*10^n-1)/gcd(k-1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n-1)/gcd(k-1,11-1) with integer n>=1 * if k < 376, then there are infinitely many primes of the form (k*12^n-1)/gcd(k-1,12-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*13^n-1)/gcd(k-1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n-1)/gcd(k-1,14-1) with integer n>=1 * if k < 622403, then there are infinitely many primes of the form (k*15^n-1)/gcd(k-1,15-1) with integer n>=1 * if k < 100, then there are infinitely many primes of the form (k*16^n-1)/gcd(k-1,16-1) with integer n>=1 * if k < 49, then there are infinitely many primes of the form (k*17^n-1)/gcd(k-1,17-1) with integer n>=1 * if k < 246, then there are infinitely many primes of the form (k*18^n-1)/gcd(k-1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n-1)/gcd(k-1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n-1)/gcd(k-1,20-1) with integer n>=1 * if k < 45, then there are infinitely many primes of the form (k*21^n-1)/gcd(k-1,21-1) with integer n>=1 * if k < 2738, then there are infinitely many primes of the form (k*22^n-1)/gcd(k-1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n-1)/gcd(k-1,23-1) with integer n>=1 * if k < 32336, then there are infinitely many primes of the form (k*24^n-1)/gcd(k-1,24-1) with integer n>=1 * if k < 105, then there are infinitely many primes of the form (k*25^n-1)/gcd(k-1,25-1) with integer n>=1 * if k < 149, then there are infinitely many primes of the form (k*26^n-1)/gcd(k-1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n-1)/gcd(k-1,27-1) with integer n>=1 * if k < 3769, then there are infinitely many primes of the form (k*28^n-1)/gcd(k-1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n-1)/gcd(k-1,29-1) with integer n>=1 * if k < 4928, then there are infinitely many primes of the form (k*30^n-1)/gcd(k-1,30-1) with integer n>=1 * if k < 145, then there are infinitely many primes of the form (k*31^n-1)/gcd(k-1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n-1)/gcd(k-1,32-1) with integer n>=1 * if k < 545, then there are infinitely many primes of the form (k*33^n-1)/gcd(k-1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n-1)/gcd(k-1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n-1)/gcd(k-1,35-1) with integer n>=1 * if k < 33791, then there are infinitely many primes of the form (k*36^n-1)/gcd(k-1,36-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*37^n-1)/gcd(k-1,37-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*38^n-1)/gcd(k-1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n-1)/gcd(k-1,39-1) with integer n>=1 * if k < 25462, then there are infinitely many primes of the form (k*40^n-1)/gcd(k-1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n-1)/gcd(k-1,41-1) with integer n>=1 * if k < 15137, then there are infinitely many primes of the form (k*42^n-1)/gcd(k-1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n-1)/gcd(k-1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n-1)/gcd(k-1,44-1) with integer n>=1 * if k < 93, then there are infinitely many primes of the form (k*45^n-1)/gcd(k-1,45-1) with integer n>=1 * if k < 928, then there are infinitely many primes of the form (k*46^n-1)/gcd(k-1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n-1)/gcd(k-1,47-1) with integer n>=1 * if k < 3226, then there are infinitely many primes of the form (k*48^n-1)/gcd(k-1,48-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*49^n-1)/gcd(k-1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n-1)/gcd(k-1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n-1)/gcd(k-1,51-1) with integer n>=1 * if k < 25015, then there are infinitely many primes of the form (k*52^n-1)/gcd(k-1,52-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*53^n-1)/gcd(k-1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n-1)/gcd(k-1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n-1)/gcd(k-1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n-1)/gcd(k-1,56-1) with integer n>=1 * if k < 144, then there are infinitely many primes of the form (k*57^n-1)/gcd(k-1,57-1) with integer n>=1 * if k < 547, then there are infinitely many primes of the form (k*58^n-1)/gcd(k-1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n-1)/gcd(k-1,59-1) with integer n>=1 * if k < 20558, then there are infinitely many primes of the form (k*60^n-1)/gcd(k-1,60-1) with integer n>=1 * if k < 125, then there are infinitely many primes of the form (k*61^n-1)/gcd(k-1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n-1)/gcd(k-1,62-1) with integer n>=1 * if k < 857, then there are infinitely many primes of the form (k*63^n-1)/gcd(k-1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n-1)/gcd(k-1,64-1) with integer n>=1
2020-06-19, 09:25   #819
sweety439

Nov 2016

5×571 Posts

Quote:
 Originally Posted by sweety439 Stronger conjectures: (assuming k is positive integer) * if k < 78557, then there are infinitely many primes of the form (k*2^n+1)/gcd(k+1,2-1) with integer n>=1 * if k < 11047, then there are infinitely many primes of the form (k*3^n+1)/gcd(k+1,3-1) with integer n>=1 * if k < 419, then there are infinitely many primes of the form (k*4^n+1)/gcd(k+1,4-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*5^n+1)/gcd(k+1,5-1) with integer n>=1 * if k < 174308, then there are infinitely many primes of the form (k*6^n+1)/gcd(k+1,6-1) with integer n>=1 * if k < 209, then there are infinitely many primes of the form (k*7^n+1)/gcd(k+1,7-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*8^n+1)/gcd(k+1,8-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*9^n+1)/gcd(k+1,9-1) with integer n>=1 * if k < 989, then there are infinitely many primes of the form (k*10^n+1)/gcd(k+1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n+1)/gcd(k+1,11-1) with integer n>=1 * if k < 521, then there are infinitely many primes of the form (k*12^n+1)/gcd(k+1,12-1) with integer n>=1 * if k < 15, then there are infinitely many primes of the form (k*13^n+1)/gcd(k+1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n+1)/gcd(k+1,14-1) with integer n>=1 * if k < 673029, then there are infinitely many primes of the form (k*15^n+1)/gcd(k+1,15-1) with integer n>=1 * if k < 38, then there are infinitely many primes of the form (k*16^n+1)/gcd(k+1,16-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*17^n+1)/gcd(k+1,17-1) with integer n>=1 * if k < 398, then there are infinitely many primes of the form (k*18^n+1)/gcd(k+1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n+1)/gcd(k+1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n+1)/gcd(k+1,20-1) with integer n>=1 * if k < 23, then there are infinitely many primes of the form (k*21^n+1)/gcd(k+1,21-1) with integer n>=1 * if k < 2253, then there are infinitely many primes of the form (k*22^n+1)/gcd(k+1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n+1)/gcd(k+1,23-1) with integer n>=1 * if k < 30651, then there are infinitely many primes of the form (k*24^n+1)/gcd(k+1,24-1) with integer n>=1 * if k < 79, then there are infinitely many primes of the form (k*25^n+1)/gcd(k+1,25-1) with integer n>=1 * if k < 221, then there are infinitely many primes of the form (k*26^n+1)/gcd(k+1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n+1)/gcd(k+1,27-1) with integer n>=1 * if k < 4554, then there are infinitely many primes of the form (k*28^n+1)/gcd(k+1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n+1)/gcd(k+1,29-1) with integer n>=1 * if k < 867, then there are infinitely many primes of the form (k*30^n+1)/gcd(k+1,30-1) with integer n>=1 * if k < 239, then there are infinitely many primes of the form (k*31^n+1)/gcd(k+1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n+1)/gcd(k+1,32-1) with integer n>=1 * if k < 511, then there are infinitely many primes of the form (k*33^n+1)/gcd(k+1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n+1)/gcd(k+1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n+1)/gcd(k+1,35-1) with integer n>=1 * if k < 1886, then there are infinitely many primes of the form (k*36^n+1)/gcd(k+1,36-1) with integer n>=1 * if k < 39, then there are infinitely many primes of the form (k*37^n+1)/gcd(k+1,37-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*38^n+1)/gcd(k+1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n+1)/gcd(k+1,39-1) with integer n>=1 * if k < 47723, then there are infinitely many primes of the form (k*40^n+1)/gcd(k+1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n+1)/gcd(k+1,41-1) with integer n>=1 * if k < 13372, then there are infinitely many primes of the form (k*42^n+1)/gcd(k+1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n+1)/gcd(k+1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n+1)/gcd(k+1,44-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*45^n+1)/gcd(k+1,45-1) with integer n>=1 * if k < 881, then there are infinitely many primes of the form (k*46^n+1)/gcd(k+1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n+1)/gcd(k+1,47-1) with integer n>=1 * if k < 1219, then there are infinitely many primes of the form (k*48^n+1)/gcd(k+1,48-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*49^n+1)/gcd(k+1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n+1)/gcd(k+1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n+1)/gcd(k+1,51-1) with integer n>=1 * if k < 28674, then there are infinitely many primes of the form (k*52^n+1)/gcd(k+1,52-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*53^n+1)/gcd(k+1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n+1)/gcd(k+1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n+1)/gcd(k+1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n+1)/gcd(k+1,56-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*57^n+1)/gcd(k+1,57-1) with integer n>=1 * if k < 488, then there are infinitely many primes of the form (k*58^n+1)/gcd(k+1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n+1)/gcd(k+1,59-1) with integer n>=1 * if k < 16957, then there are infinitely many primes of the form (k*60^n+1)/gcd(k+1,60-1) with integer n>=1 * if k < 63, then there are infinitely many primes of the form (k*61^n+1)/gcd(k+1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n+1)/gcd(k+1,62-1) with integer n>=1 * if k < 1589, then there are infinitely many primes of the form (k*63^n+1)/gcd(k+1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n+1)/gcd(k+1,64-1) with integer n>=1 * if k < 509203, then there are infinitely many primes of the form (k*2^n-1)/gcd(k-1,2-1) with integer n>=1 * if k < 12119, then there are infinitely many primes of the form (k*3^n-1)/gcd(k-1,3-1) with integer n>=1 * if k < 361, then there are infinitely many primes of the form (k*4^n-1)/gcd(k-1,4-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*5^n-1)/gcd(k-1,5-1) with integer n>=1 * if k < 84687, then there are infinitely many primes of the form (k*6^n-1)/gcd(k-1,6-1) with integer n>=1 * if k < 457, then there are infinitely many primes of the form (k*7^n-1)/gcd(k-1,7-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*8^n-1)/gcd(k-1,8-1) with integer n>=1 * if k < 41, then there are infinitely many primes of the form (k*9^n-1)/gcd(k-1,9-1) with integer n>=1 * if k < 334, then there are infinitely many primes of the form (k*10^n-1)/gcd(k-1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n-1)/gcd(k-1,11-1) with integer n>=1 * if k < 376, then there are infinitely many primes of the form (k*12^n-1)/gcd(k-1,12-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*13^n-1)/gcd(k-1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n-1)/gcd(k-1,14-1) with integer n>=1 * if k < 622403, then there are infinitely many primes of the form (k*15^n-1)/gcd(k-1,15-1) with integer n>=1 * if k < 100, then there are infinitely many primes of the form (k*16^n-1)/gcd(k-1,16-1) with integer n>=1 * if k < 49, then there are infinitely many primes of the form (k*17^n-1)/gcd(k-1,17-1) with integer n>=1 * if k < 246, then there are infinitely many primes of the form (k*18^n-1)/gcd(k-1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n-1)/gcd(k-1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n-1)/gcd(k-1,20-1) with integer n>=1 * if k < 45, then there are infinitely many primes of the form (k*21^n-1)/gcd(k-1,21-1) with integer n>=1 * if k < 2738, then there are infinitely many primes of the form (k*22^n-1)/gcd(k-1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n-1)/gcd(k-1,23-1) with integer n>=1 * if k < 32336, then there are infinitely many primes of the form (k*24^n-1)/gcd(k-1,24-1) with integer n>=1 * if k < 105, then there are infinitely many primes of the form (k*25^n-1)/gcd(k-1,25-1) with integer n>=1 * if k < 149, then there are infinitely many primes of the form (k*26^n-1)/gcd(k-1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n-1)/gcd(k-1,27-1) with integer n>=1 * if k < 3769, then there are infinitely many primes of the form (k*28^n-1)/gcd(k-1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n-1)/gcd(k-1,29-1) with integer n>=1 * if k < 4928, then there are infinitely many primes of the form (k*30^n-1)/gcd(k-1,30-1) with integer n>=1 * if k < 145, then there are infinitely many primes of the form (k*31^n-1)/gcd(k-1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n-1)/gcd(k-1,32-1) with integer n>=1 * if k < 545, then there are infinitely many primes of the form (k*33^n-1)/gcd(k-1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n-1)/gcd(k-1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n-1)/gcd(k-1,35-1) with integer n>=1 * if k < 33791, then there are infinitely many primes of the form (k*36^n-1)/gcd(k-1,36-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*37^n-1)/gcd(k-1,37-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*38^n-1)/gcd(k-1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n-1)/gcd(k-1,39-1) with integer n>=1 * if k < 25462, then there are infinitely many primes of the form (k*40^n-1)/gcd(k-1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n-1)/gcd(k-1,41-1) with integer n>=1 * if k < 15137, then there are infinitely many primes of the form (k*42^n-1)/gcd(k-1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n-1)/gcd(k-1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n-1)/gcd(k-1,44-1) with integer n>=1 * if k < 93, then there are infinitely many primes of the form (k*45^n-1)/gcd(k-1,45-1) with integer n>=1 * if k < 928, then there are infinitely many primes of the form (k*46^n-1)/gcd(k-1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n-1)/gcd(k-1,47-1) with integer n>=1 * if k < 3226, then there are infinitely many primes of the form (k*48^n-1)/gcd(k-1,48-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*49^n-1)/gcd(k-1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n-1)/gcd(k-1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n-1)/gcd(k-1,51-1) with integer n>=1 * if k < 25015, then there are infinitely many primes of the form (k*52^n-1)/gcd(k-1,52-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*53^n-1)/gcd(k-1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n-1)/gcd(k-1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n-1)/gcd(k-1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n-1)/gcd(k-1,56-1) with integer n>=1 * if k < 144, then there are infinitely many primes of the form (k*57^n-1)/gcd(k-1,57-1) with integer n>=1 * if k < 547, then there are infinitely many primes of the form (k*58^n-1)/gcd(k-1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n-1)/gcd(k-1,59-1) with integer n>=1 * if k < 20558, then there are infinitely many primes of the form (k*60^n-1)/gcd(k-1,60-1) with integer n>=1 * if k < 125, then there are infinitely many primes of the form (k*61^n-1)/gcd(k-1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n-1)/gcd(k-1,62-1) with integer n>=1 * if k < 857, then there are infinitely many primes of the form (k*63^n-1)/gcd(k-1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n-1)/gcd(k-1,64-1) with integer n>=1
Some k's have algebra factors, so there are additional conditions for some conjectures:

* For (k*8^n+1)/gcd(k+1,8-1), k is not cube of integer
* For (k*16^n+1)/gcd(k+1,16-1), k is not of the form 4*q^4 with integer q
* For (k*27^n+1)/gcd(k+1,27-1), k is not cube of integer
* For (k*32^n+1)/gcd(k+1,32-1), k is not fifth power of integer
* For (k*64^n+1)/gcd(k+1,64-1), k is not cube of integer

* For (k*4^n-1)/gcd(k-1,4-1), k is not square of integer
* For (k*8^n-1)/gcd(k-1,8-1), k is not cube of integer
* For (k*9^n-1)/gcd(k-1,9-1), k is not square of integer
* For (k*12^n-1)/gcd(k-1,12-1), k is not of the form m^2 with integer m == 5 or 8 mod 13 nor of the form 3*m^2 with integer m == 3 or 10 mod 13
* For (k*16^n-1)/gcd(k-1,16-1), k is not square of integer
* For (k*19^n-1)/gcd(k-1,19-1), k is not of the form m^2 with integer m == 2 or 3 mod 5
* For (k*24^n-1)/gcd(k-1,24-1), k is not of the form m^2 with integer m == 2 or 3 mod 5 nor of the form 6*m^2 with integer m == 1 or 4 mod 5
* For (k*25^n-1)/gcd(k-1,25-1), k is not square of integer
* For (k*27^n-1)/gcd(k-1,27-1), k is not cube of integer
* For (k*28^n-1)/gcd(k-1,28-1), k is not of the form m^2 with integer m == 12 or 17 mod 29 nor of the form 7*m^2 with integer m == 5 or 24 mod 29
* For (k*30^n-1)/gcd(k-1,30-1), k is not equal to 1369
* For (k*32^n-1)/gcd(k-1,32-1), k is not fifth power of integer
* For (k*33^n-1)/gcd(k-1,33-1), k is not of the form m^2 with integer m == 4 or 13 mod 17 nor of the form 33*m^2 with integer m == 4 or 13 mod 17 nor of the form m^2 with integer m == 15 or 17 mod 32
* For (k*34^n-1)/gcd(k-1,34-1), k is not of the form m^2 with integer m == 2 or 3 mod 5
* For (k*36^n-1)/gcd(k-1,36-1), k is not square of integer
* For (k*39^n-1)/gcd(k-1,39-1), k is not of the form m^2 with integer m == 2 or 3 mod 5
* For (k*40^n-1)/gcd(k-1,40-1), k is not of the form m^2 with integer m == 9 or 32 mod 41 nor of the form 10*m^2 with integer m == 18 or 23 mod 41
* For (k*49^n-1)/gcd(k-1,49-1), k is not square of integer
* For (k*52^n-1)/gcd(k-1,52-1), k is not of the form m^2 with integer m == 23 or 30 mod 53 nor of the form 13*m^2 with integer m == 7 or 46 mod 53
* For (k*54^n-1)/gcd(k-1,54-1), k is not of the form m^2 with integer m == 2 or 3 mod 5 nor of the form 6*m^2 with integer m == 1 or 4 mod 5
* For (k*57^n-1)/gcd(k-1,57-1), k is not of the form m^2 with integer m == 3 or 5 mod 8
* For (k*60^n-1)/gcd(k-1,60-1), k is not of the form m^2 with integer m == 11 or 50 mod 61 nor of the form 15*m^2 with integer m == 22 or 39 mod 61
* For (k*64^n-1)/gcd(k-1,64-1), k is not square of integer nor cube of integer

Last fiddled with by sweety439 on 2020-06-19 at 17:52

2020-06-19, 17:02   #820
sweety439

Nov 2016

5·571 Posts

Quote:
 Originally Posted by sweety439 Stronger conjectures: (assuming k is positive integer) * if k < 78557, then there are infinitely many primes of the form (k*2^n+1)/gcd(k+1,2-1) with integer n>=1 * if k < 11047, then there are infinitely many primes of the form (k*3^n+1)/gcd(k+1,3-1) with integer n>=1 * if k < 419, then there are infinitely many primes of the form (k*4^n+1)/gcd(k+1,4-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*5^n+1)/gcd(k+1,5-1) with integer n>=1 * if k < 174308, then there are infinitely many primes of the form (k*6^n+1)/gcd(k+1,6-1) with integer n>=1 * if k < 209, then there are infinitely many primes of the form (k*7^n+1)/gcd(k+1,7-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*8^n+1)/gcd(k+1,8-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*9^n+1)/gcd(k+1,9-1) with integer n>=1 * if k < 989, then there are infinitely many primes of the form (k*10^n+1)/gcd(k+1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n+1)/gcd(k+1,11-1) with integer n>=1 * if k < 521, then there are infinitely many primes of the form (k*12^n+1)/gcd(k+1,12-1) with integer n>=1 * if k < 15, then there are infinitely many primes of the form (k*13^n+1)/gcd(k+1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n+1)/gcd(k+1,14-1) with integer n>=1 * if k < 673029, then there are infinitely many primes of the form (k*15^n+1)/gcd(k+1,15-1) with integer n>=1 * if k < 38, then there are infinitely many primes of the form (k*16^n+1)/gcd(k+1,16-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*17^n+1)/gcd(k+1,17-1) with integer n>=1 * if k < 398, then there are infinitely many primes of the form (k*18^n+1)/gcd(k+1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n+1)/gcd(k+1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n+1)/gcd(k+1,20-1) with integer n>=1 * if k < 23, then there are infinitely many primes of the form (k*21^n+1)/gcd(k+1,21-1) with integer n>=1 * if k < 2253, then there are infinitely many primes of the form (k*22^n+1)/gcd(k+1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n+1)/gcd(k+1,23-1) with integer n>=1 * if k < 30651, then there are infinitely many primes of the form (k*24^n+1)/gcd(k+1,24-1) with integer n>=1 * if k < 79, then there are infinitely many primes of the form (k*25^n+1)/gcd(k+1,25-1) with integer n>=1 * if k < 221, then there are infinitely many primes of the form (k*26^n+1)/gcd(k+1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n+1)/gcd(k+1,27-1) with integer n>=1 * if k < 4554, then there are infinitely many primes of the form (k*28^n+1)/gcd(k+1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n+1)/gcd(k+1,29-1) with integer n>=1 * if k < 867, then there are infinitely many primes of the form (k*30^n+1)/gcd(k+1,30-1) with integer n>=1 * if k < 239, then there are infinitely many primes of the form (k*31^n+1)/gcd(k+1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n+1)/gcd(k+1,32-1) with integer n>=1 * if k < 511, then there are infinitely many primes of the form (k*33^n+1)/gcd(k+1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n+1)/gcd(k+1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n+1)/gcd(k+1,35-1) with integer n>=1 * if k < 1886, then there are infinitely many primes of the form (k*36^n+1)/gcd(k+1,36-1) with integer n>=1 * if k < 39, then there are infinitely many primes of the form (k*37^n+1)/gcd(k+1,37-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*38^n+1)/gcd(k+1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n+1)/gcd(k+1,39-1) with integer n>=1 * if k < 47723, then there are infinitely many primes of the form (k*40^n+1)/gcd(k+1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n+1)/gcd(k+1,41-1) with integer n>=1 * if k < 13372, then there are infinitely many primes of the form (k*42^n+1)/gcd(k+1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n+1)/gcd(k+1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n+1)/gcd(k+1,44-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*45^n+1)/gcd(k+1,45-1) with integer n>=1 * if k < 881, then there are infinitely many primes of the form (k*46^n+1)/gcd(k+1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n+1)/gcd(k+1,47-1) with integer n>=1 * if k < 1219, then there are infinitely many primes of the form (k*48^n+1)/gcd(k+1,48-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*49^n+1)/gcd(k+1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n+1)/gcd(k+1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n+1)/gcd(k+1,51-1) with integer n>=1 * if k < 28674, then there are infinitely many primes of the form (k*52^n+1)/gcd(k+1,52-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*53^n+1)/gcd(k+1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n+1)/gcd(k+1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n+1)/gcd(k+1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n+1)/gcd(k+1,56-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*57^n+1)/gcd(k+1,57-1) with integer n>=1 * if k < 488, then there are infinitely many primes of the form (k*58^n+1)/gcd(k+1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n+1)/gcd(k+1,59-1) with integer n>=1 * if k < 16957, then there are infinitely many primes of the form (k*60^n+1)/gcd(k+1,60-1) with integer n>=1 * if k < 63, then there are infinitely many primes of the form (k*61^n+1)/gcd(k+1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n+1)/gcd(k+1,62-1) with integer n>=1 * if k < 1589, then there are infinitely many primes of the form (k*63^n+1)/gcd(k+1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n+1)/gcd(k+1,64-1) with integer n>=1 * if k < 509203, then there are infinitely many primes of the form (k*2^n-1)/gcd(k-1,2-1) with integer n>=1 * if k < 12119, then there are infinitely many primes of the form (k*3^n-1)/gcd(k-1,3-1) with integer n>=1 * if k < 361, then there are infinitely many primes of the form (k*4^n-1)/gcd(k-1,4-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*5^n-1)/gcd(k-1,5-1) with integer n>=1 * if k < 84687, then there are infinitely many primes of the form (k*6^n-1)/gcd(k-1,6-1) with integer n>=1 * if k < 457, then there are infinitely many primes of the form (k*7^n-1)/gcd(k-1,7-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*8^n-1)/gcd(k-1,8-1) with integer n>=1 * if k < 41, then there are infinitely many primes of the form (k*9^n-1)/gcd(k-1,9-1) with integer n>=1 * if k < 334, then there are infinitely many primes of the form (k*10^n-1)/gcd(k-1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n-1)/gcd(k-1,11-1) with integer n>=1 * if k < 376, then there are infinitely many primes of the form (k*12^n-1)/gcd(k-1,12-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*13^n-1)/gcd(k-1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n-1)/gcd(k-1,14-1) with integer n>=1 * if k < 622403, then there are infinitely many primes of the form (k*15^n-1)/gcd(k-1,15-1) with integer n>=1 * if k < 100, then there are infinitely many primes of the form (k*16^n-1)/gcd(k-1,16-1) with integer n>=1 * if k < 49, then there are infinitely many primes of the form (k*17^n-1)/gcd(k-1,17-1) with integer n>=1 * if k < 246, then there are infinitely many primes of the form (k*18^n-1)/gcd(k-1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n-1)/gcd(k-1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n-1)/gcd(k-1,20-1) with integer n>=1 * if k < 45, then there are infinitely many primes of the form (k*21^n-1)/gcd(k-1,21-1) with integer n>=1 * if k < 2738, then there are infinitely many primes of the form (k*22^n-1)/gcd(k-1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n-1)/gcd(k-1,23-1) with integer n>=1 * if k < 32336, then there are infinitely many primes of the form (k*24^n-1)/gcd(k-1,24-1) with integer n>=1 * if k < 105, then there are infinitely many primes of the form (k*25^n-1)/gcd(k-1,25-1) with integer n>=1 * if k < 149, then there are infinitely many primes of the form (k*26^n-1)/gcd(k-1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n-1)/gcd(k-1,27-1) with integer n>=1 * if k < 3769, then there are infinitely many primes of the form (k*28^n-1)/gcd(k-1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n-1)/gcd(k-1,29-1) with integer n>=1 * if k < 4928, then there are infinitely many primes of the form (k*30^n-1)/gcd(k-1,30-1) with integer n>=1 * if k < 145, then there are infinitely many primes of the form (k*31^n-1)/gcd(k-1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n-1)/gcd(k-1,32-1) with integer n>=1 * if k < 545, then there are infinitely many primes of the form (k*33^n-1)/gcd(k-1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n-1)/gcd(k-1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n-1)/gcd(k-1,35-1) with integer n>=1 * if k < 33791, then there are infinitely many primes of the form (k*36^n-1)/gcd(k-1,36-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*37^n-1)/gcd(k-1,37-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*38^n-1)/gcd(k-1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n-1)/gcd(k-1,39-1) with integer n>=1 * if k < 25462, then there are infinitely many primes of the form (k*40^n-1)/gcd(k-1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n-1)/gcd(k-1,41-1) with integer n>=1 * if k < 15137, then there are infinitely many primes of the form (k*42^n-1)/gcd(k-1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n-1)/gcd(k-1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n-1)/gcd(k-1,44-1) with integer n>=1 * if k < 93, then there are infinitely many primes of the form (k*45^n-1)/gcd(k-1,45-1) with integer n>=1 * if k < 928, then there are infinitely many primes of the form (k*46^n-1)/gcd(k-1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n-1)/gcd(k-1,47-1) with integer n>=1 * if k < 3226, then there are infinitely many primes of the form (k*48^n-1)/gcd(k-1,48-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*49^n-1)/gcd(k-1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n-1)/gcd(k-1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n-1)/gcd(k-1,51-1) with integer n>=1 * if k < 25015, then there are infinitely many primes of the form (k*52^n-1)/gcd(k-1,52-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*53^n-1)/gcd(k-1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n-1)/gcd(k-1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n-1)/gcd(k-1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n-1)/gcd(k-1,56-1) with integer n>=1 * if k < 144, then there are infinitely many primes of the form (k*57^n-1)/gcd(k-1,57-1) with integer n>=1 * if k < 547, then there are infinitely many primes of the form (k*58^n-1)/gcd(k-1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n-1)/gcd(k-1,59-1) with integer n>=1 * if k < 20558, then there are infinitely many primes of the form (k*60^n-1)/gcd(k-1,60-1) with integer n>=1 * if k < 125, then there are infinitely many primes of the form (k*61^n-1)/gcd(k-1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n-1)/gcd(k-1,62-1) with integer n>=1 * if k < 857, then there are infinitely many primes of the form (k*63^n-1)/gcd(k-1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n-1)/gcd(k-1,64-1) with integer n>=1
We can also make much stronger conjectures (the 1st, 2nd, 3rd, and 4th Sierpinski/Riesel conjectures):

If k < 4th CK and does not equal to 1st CK, 2nd CK, or 3rd CK, then there are infinitely many primes of the form (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) with integer n>=1

Sierpinski:

Code:
b: 1st CK, 2nd CK, 3rd CK, 4th CK
2: 78557, 157114, 271129, 271577,
3: 11047, 23789, 27221, 32549,
4: 419, 659, 794, 1466,
5: 7, 11, 31, 35,
6: 174308, 188299, 243417, 282001,
7: 209, 1463, 3305, 3533,
8: 47, 79, 83, 181,
9: 31, 39, 111, 119,
10: 989, 1121, 3653, 3662,
11: 5, 7, 17, 19,
12: 521, 597, 1143, 1509,
13: 15, 27, 47, 71,
14: 4, 11, 19, 26,
15: 673029, 2105431, 2692337, 4621459,
16: 38, 194, 524, 608,
17: 31, 47, 127, 143,
18: 398, 512, 571, 989,
19: 9, 11, 29, 31,
20: 8, 13, 29, 34,
21: 23, 43, 47, 111,
22: 2253, 4946, 6694, 8417,
23: 5, 7, 17, 19,
24: 30651, 66356, 77554, 84766,
25: 79, 103, 185, 287,
26: 221, 284, 1627, 1766,
27: 13, 15, 41, 43,
28: 4554, 8293, 13687, 18996,
29: 4, 7, 11, 19,
30: 867, 9859, 10386, 10570,
31: 239, 293, 521, 1025,
32: 10, 23, 43, 56,
33: 511, 543, 1599, 1631,
34: 6, 29, 41, 64,
35: 5, 7, 17, 19,
36: 1886, 11093, 67896, 123189,
37: 39, 75, 87, 191,
38: 14, 16, 25, 53,
39: 9, 11, 29, 31,
40: 47723, 67241, 68963, 133538,
41: 8, 13, 15, 23,
42: 13372, 30359, 47301, 60758,
43: 21, 23, 65, 67,
44: 4, 11, 19, 26,
45: 47, 91, 231, 275,
46: 881, 1592, 2519, 3104,
47: 5, 7, 8, 16,
48: 1219, 3403, 5531, 5613,
49: 31, 79, 179, 191,
50: 16, 35, 67, 86,
51: 25, 27, 77, 79,
52: 28674, 57398, 83262, 117396,
53: 7, 11, 31, 35,
54: 21, 34, 76, 89,
55: 13, 15, 41, 43,
56: 20, 37, 77, 94,
57: 47, 175, 231, 311,
58: 488, 1592, 7766, 8312,
59: 4, 5, 7, 9,
60: 16957, 84486, 138776, 199103,
61: 63, 123, 311, 371,
62: 8, 13, 29, 34,
63: 1589, 2381, 4827, 7083,
64: 14, 51, 79, 116,
65: 10, 23, 43, 56,
66:
67: 26, 33, 35, 101,
68: 22, 36, 47, 56,
69: 6, 15, 19, 27,
70: 11077, 20591, 22719, 25914,
71: 5, 7, 17, 19,
72: 731, 1313, 1461, 3724,
73: 47, 223, 255, 295,
74: 4, 11, 19, 26,
75: 37, 39, 113, 115,
76: 34, 43, 111, 120,
77: 7, 11, 14, 25,
78: 96144, 186123, 288507, 390656,
79: 9, 11, 29, 31,
80: 1039, 3181, 7438, 12211,
81: 575, 649, 655, 1167,
82: 19587, 29051, 37847, 46149,
83: 5, 7, 8, 13,
84: 16, 69, 101, 154,
85: 87, 171, 431, 515,
86: 28, 59, 115, 146,
87: 21, 23, 65, 67,
88: 26, 179, 311, 521,
89: 4, 11, 19, 23,
90: 27, 64, 118, 155,
91: 45, 47, 137, 139,
92: 32, 61, 125, 154,
93: 95, 187, 471, 563,
94: 39, 56, 134, 151,
95: 5, 7, 17, 19,
96: 68869, 353081, 426217, 427383,
97: 127, 223, 575, 671,
98: 10, 16, 23, 38,
99: 9, 11, 29, 31,
100: 62, 233, 332, 836,
101: 7, 11, 16, 31,
102: 293, 1342, 6060, 6240,
103: 25, 27, 77, 79,
104: 4, 6, 8, 11,
105: 319, 423, 1167, 1271,
106: 2387, 5480, 14819, 17207,
107: 5, 7, 17, 19,
108: 26270, 102677, 131564, 132872,
109: 19, 21, 23, 31,
110: 38, 73, 149, 184,
111: 13, 15, 41, 43,
112: 2261, 2939, 3502, 5988,
113: 20, 31, 37, 47,
114: 24, 91, 139, 206,
115: 57, 59, 173, 175,
116: 14, 25, 53, 64,
117: 119, 235, 327, 591,
118: 50, 69, 169, 188,
119: 4, 5, 7, 9,
120:
121: 27, 103, 110, 293,
122: 40, 47, 79, 83,
123: 55, 61, 63, 69,
124: 31001, 56531, 77381, 145994,
125: 7, 8, 11, 13,
126: 766700, 1835532, 2781934, 2986533,
127: 6343, 7909, 12923, 13701,
128: 44, 85, 98, 173,
129: 14, 51, 79, 116,
130: 1049, 2432, 7073, 9602,
131: 5, 7, 10, 17,
132: 13, 20, 113, 153,
133: 59, 135, 267, 671,
134: 4, 11, 19, 26,
135: 33, 35, 101, 103,
136: 29180, 90693, 151660, 243037,
137: 22, 23, 31, 47,
138: 2781, 3752, 4308, 7229,
139: 6, 9, 11, 13,
140: 46, 95, 187, 236,
141: 143, 283, 711, 851,
142: 12, 131, 155, 221,
143: 5, 7, 17, 19,
144: 59, 86, 204, 231,
145: 1023, 1167, 2159, 2367,
146: 8, 13, 29, 34,
147: 73, 75, 221, 223,
148: 3128, 4022, 4471, 7749,
149: 4, 7, 11, 19,
150: 49074, 95733, 539673, 611098,
151: 37, 39, 113, 115,
152: 16, 35, 67, 86,
153: 15, 34, 43, 55,
154: 61, 94, 216, 249,
155: 5, 7, 14, 17,
156:
157: 47, 59, 159, 191,
158: 52, 107, 122, 211,
159: 9, 11, 29, 31,
160: 22, 139, 183, 300,
161: 95, 127, 287, 319,
162: 6193, 6682, 7336, 14343,
163: 81, 83, 245, 247,
164: 4, 10, 11, 19,
165: 167, 331, 831, 995,
166: 335, 5510, 7349, 9854,
167: 5, 7, 8, 13,
168: 9244, 9658, 15638, 20357,
169: 16, 31, 39, 69,
170: 20, 37, 77, 94,
171: 85, 87, 257, 259,
172: 62, 108, 836, 1070,
173: 7, 11, 28, 31,
174: 6, 29, 41, 64,
175: 21, 23, 65, 67,
176: 58, 119, 235, 296,
177: 79, 447, 1247, 1423,
178: 569, 797, 953, 1031,
179: 4, 5, 7, 9,
180: 1679679,
181: 15, 27, 51, 64,
182: 23, 62, 121, 211,
183: 45, 47, 69, 101,
184: 36, 149, 221, 269,
185: 23, 31, 32, 61,
186: 67, 120, 254, 307,
187: 47, 83, 93, 95,
188: 8, 13, 29, 34,
189: 19, 31, 39, 56,
190: 2157728, 3146151, 3713039, 4352889,
191: 5, 7, 17, 19,
192: 7879, 8686, 17371, 19494,
193: 2687, 6015, 6207, 9343,
194: 4, 11, 14, 19,
195: 13, 15, 41, 43,
196: 16457, 78689, 86285, 95147,
197: 7, 10, 11, 23,
198: 4105, 19484, 21649, 23581,
199: 9, 11, 29, 31,
200: 47, 68, 103, 118,
201: 607, 807, 2223, 2423,
202: 57, 146, 260, 349,
203: 5, 7, 16, 17,
204: 81, 124, 286, 329,
205: 207, 411, 1031, 1235,
206: 22, 47, 91, 116,
207: 25, 27, 77, 79,
208: 56, 98, 153, 265,
209: 4, 6, 8, 11,
210:
211: 105, 107, 317, 319,
212: 70, 143, 283, 285,
213: 51, 215, 339, 427,
214: 44, 171, 236, 259,
215: 5, 7, 17, 19,
216: 92, 125, 309, 342,
217: 655, 863, 871, 919,
218: 74, 145, 293, 364,
219: 9, 11, 21, 23,
220: 50, 103, 118, 324,
221: 7, 11, 31, 35,
222: 333163, 352341, 389359, 410098,
223: 13, 15, 41, 43,
224: 4, 11, 19, 26,
225: 3391, 3615, 10623, 10847,
226: 2915, 11744, 12563, 15704,
227: 5, 7, 17, 19,
228: 1146, 7098, 8474, 25647,
229: 19, 24, 31, 47,
230: 8, 10, 13, 23,
231: 57, 59, 173, 175,
232: 2564, 18992, 27527, 46520,
233: 14, 23, 25, 31,
234: 46, 189, 281, 424,
235: 107, 117, 119, 255,
236: 80, 157, 317, 394,
237: 15, 27, 50, 67,
238: 34571, 36746, 42449, 48038,
239: 4, 5, 7, 9,
240: 1722187, 1933783, 2799214,
241: 175, 287, 527, 639,
242: 8, 16, 38, 47,
243: 121, 123, 285, 365,
244: 6, 29, 41, 64,
245: 7, 11, 31, 35,
246: 77, 170, 324, 417,
247: 61, 63, 185, 187,
248: 82, 167, 331, 416,
249: 31, 39, 111, 119,
250: 9788, 23885, 33539, 50450,
251: 5, 7, 8, 13,
252: 45, 116, 144, 208,
253: 255, 327, 507, 691,
254: 4, 11, 16, 19,
255: 245, 365, 493, 499,
256: 38, 194, 467, 524,
Riesel:

Code:
b: 1st CK, 2nd CK, 3rd CK, 4th CK
2: 509203, 762701, 777149, 784109,
3: 12119, 20731, 21997, 28297,
4: 361, 919, 1114, 1444,
5: 13, 17, 37, 41,
6: 84687, 133946, 176602, 213410,
7: 457, 1291, 3199, 3313,
8: 14, 112, 116, 148,
9: 41, 49, 74, 121,
10: 334, 1585, 1882, 3340,
11: 5, 7, 17, 19,
12: 376, 742, 1288, 1364,
13: 29, 41, 69, 85,
14: 4, 11, 19, 26,
15: 622403, 1346041, 2742963,
16: 100, 172, 211, 295,
17: 49, 59, 65, 86,
18: 246, 664, 723, 837,
19: 9, 11, 29, 31,
20: 8, 13, 29, 34,
21: 45, 65, 133, 153,
22: 2738, 4461, 6209, 8902,
23: 5, 7, 17, 19,
24: 32336, 69691, 109054, 124031,
25: 105, 129, 211, 313,
26: 149, 334, 1892, 1987,
27: 13, 15, 41, 43,
28: 3769, 9078, 14472, 18211,
29: 4, 9, 11, 13,
30: 4928, 5331, 7968, 8958,
31: 145, 265, 443, 493,
32: 10, 23, 43, 56,
33: 545, 577, 764, 1633,
34: 6, 29, 41, 64,
35: 5, 7, 17, 19,
36: 33791, 79551, 89398, 116364,
37: 29, 77, 113, 163,
38: 13, 14, 25, 53,
39: 9, 11, 29, 31,
40: 25462, 29437, 38539, 52891,
41: 8, 13, 17, 25,
42: 15137, 28594, 45536, 62523,
43: 21, 23, 65, 67,
44: 4, 11, 19, 26,
45: 93, 137, 277, 321,
46: 928, 3754, 4078, 4636,
47: 5, 7, 13, 14,
48: 3226, 4208, 7029, 7965,
49: 81, 129, 229, 241,
50: 16, 35, 67, 86,
51: 25, 27, 77, 79,
52: 25015, 25969, 35299, 60103,
53: 13, 17, 37, 41,
54: 21, 34, 76, 89,
55: 13, 15, 41, 43,
56: 20, 37, 77, 94,
57: 144, 177, 233, 289,
58: 547, 919, 1408, 1957,
59: 4, 5, 7, 9,
60: 20558, 80885, 135175, 202704,
61: 125, 185, 373, 433,
62: 8, 13, 29, 34,
63: 857, 3113, 5559, 6351,
64: 14, 51, 79, 116,
65: 10, 23, 43, 56,
66:
67: 33, 35, 37, 101,
68: 22, 43, 47, 61,
69: 6, 9, 21, 29,
70: 853, 4048, 6176, 15690,
71: 5, 7, 17, 19,
72: 293, 2481, 3722, 4744,
73: 112, 177, 297, 329,
74: 4, 11, 19, 26,
75: 37, 39, 113, 115,
76: 34, 43, 111, 120,
77: 13, 14, 17, 25,
78: 90059, 192208, 294592, 384571,
79: 9, 11, 29, 31,
80: 253, 1037, 6148, 11765,
81: 74, 575, 657, 737,
82: 22326, 36438, 44572, 64905,
83: 5, 7, 8, 13,
84: 16, 69, 101, 154,
85: 173, 257, 517, 601,
86: 28, 59, 115, 146,
87: 21, 23, 65, 67,
88: 571, 862, 898, 961,
89: 4, 11, 17, 19,
90: 27, 64, 118, 155,
91: 45, 47, 137, 139,
92: 32, 61, 125, 154,
93: 189, 281, 565, 612,
94: 39, 56, 134, 151,
95: 5, 7, 17, 19,
96: 38995, 78086, 343864, 540968,
97: 43, 225, 321, 673,
98: 10, 23, 43, 56,
99: 9, 11, 29, 31,
100: 211, 235, 334, 750,
101: 13, 16, 17, 33,
102: 1635, 1793, 4267, 4447,
103: 25, 27, 77, 79,
104: 4, 6, 8, 11,
105: 297, 425, 529, 1273,
106: 13624, 14926, 16822, 19210,
107: 5, 7, 17, 19,
108: 13406, 26270, 43601, 103835,
109: 9, 21, 34, 45,
110: 38, 73, 149, 184,
111: 13, 15, 41, 43,
112: 1357, 3843, 4406, 5084,
113: 20, 37, 49, 65,
114: 24, 91, 139, 206,
115: 57, 59, 173, 175,
116: 14, 25, 53, 64,
117: 149, 221, 237, 353,
118: 50, 69, 169, 188,
119: 4, 5, 7, 9,
120:
121: 100, 163, 211, 232,
122: 14, 40, 83, 112,
123: 13, 61, 63, 154,
124: 92881, 104716, 124009, 170386,
125: 8, 13, 17, 29,
126: 480821, 2767077, 3925190,
127: 2593, 3251, 3353, 6451,
128: 44, 59, 85, 86,
129: 14, 51, 79, 116,
130: 2563, 5896, 11134, 26632,
131: 5, 7, 10, 17,
132: 20, 69, 113, 153,
133: 17, 233, 269, 273,
134: 4, 11, 19, 26,
135: 33, 35, 101, 103,
136: 22195, 47677, 90693, 151660,
137: 17, 22, 25, 47,
138: 1806, 4727, 5283, 6254,
139: 6, 9, 11, 13,
140: 46, 95, 187, 236,
141: 285, 425, 853, 993,
142: 12, 131, 155, 219,
143: 5, 7, 17, 19,
144: 59, 86, 204, 231,
145: 1169, 1313, 3505, 3649,
146: 8, 13, 29, 34,
147: 73, 75, 221, 223,
148: 1936, 5214, 5663, 6557,
149: 4, 9, 11, 13,
150: 49074, 95733, 228764, 539673,
151: 37, 39, 113, 115,
152: 16, 35, 67, 86,
153: 34, 43, 57, 65,
154: 61, 94, 216, 249,
155: 5, 7, 14, 17,
156:
157: 17, 69, 101, 217,
158: 52, 107, 211, 266,
159: 9, 11, 29, 31,
160: 22, 139, 183, 253,
161: 65, 97, 257, 289,
162: 3259, 4726, 9292, 16299,
163: 81, 83, 245, 247,
164: 4, 10, 11, 19,
165: 79, 333, 497, 646,
166: 4174, 9019, 11023, 15532,
167: 5, 7, 8, 13,
168: 4744, 14676, 15393, 20827,
169: 16, 33, 41, 49,
170: 20, 37, 77, 94,
171: 85, 87, 257, 259,
172: 235, 982, 1108, 1171,
173: 13, 17, 28, 37,
174: 6, 21, 29, 41,
175: 21, 23, 65, 67,
176: 58, 119, 235, 296,
177: 209, 268, 577, 1156,
178: 22, 79, 87, 334,
179: 4, 5, 7, 9,
180:
181: 25, 27, 29, 41,
182: 62, 121, 245, 304,
183: 45, 47, 137, 139,
184: 36, 149, 221, 334,
185: 17, 25, 32, 61,
186: 67, 120, 254, 307,
187: 51, 79, 93, 95,
188: 8, 13, 29, 34,
189: 9, 21, 39, 49,
190: 626861, 2121627, 3182252, 3749140,
191: 5, 7, 17, 19,
192: 13897, 19492, 20459, 22968,
193: 484, 5350, 6209, 6401,
194: 4, 11, 14, 19,
195: 13, 15, 41, 43,
196: 1267, 16654, 17920, 20692,
197: 10, 13, 17, 23,
198: 3662, 8425, 10546, 13224,
199: 9, 11, 29, 31,
200: 68, 133, 268, 269,
201: 809, 1009, 2425, 2625,
202: 57, 146, 260, 349,
203: 5, 7, 14, 16,
204: 81, 124, 286, 329,
205: 25, 361, 413, 617,
206: 22, 47, 91, 116,
207: 25, 27, 77, 79,
208: 56, 153, 186, 265,
209: 4, 6, 8, 11,
210:
211: 100, 105, 107, 317,
212: 70, 143, 149, 179,
213: 57, 73, 181, 429,
214: 44, 171, 259, 386,
215: 5, 7, 17, 19,
216: 92, 125, 309, 342,
217: 337, 353, 409, 441,
218: 74, 145, 293, 364,
219: 9, 11, 21, 23,
220: 103, 118, 324, 339,
221: 13, 17, 37, 38,
222: 88530, 90091, 282094, 514016,
223: 13, 15, 41, 43,
224: 4, 11, 19, 26,
225: 3617, 3841, 10849, 11073,
226: 820, 12790, 50257, 53398,
227: 5, 7, 17, 19,
228: 16718, 33891, 35267, 41219,
229: 9, 21, 24, 49,
230: 8, 10, 13, 23,
231: 57, 59, 173, 175,
232: 27760, 72817, 98791, 100576,
233: 14, 17, 25, 53,
234: 46, 189, 281, 424,
235: 64, 117, 119, 172,
236: 80, 157, 317, 394,
237: 29, 33, 41, 50,
238: 17926, 34810, 93628, 99094,
239: 4, 5, 7, 9,
240: 2952972, 2985025, 3695736, 4812046,
241: 65, 177, 417, 529,
242: 14, 73, 101, 116,
243: 121, 123, 365, 367,
244: 6, 29, 41, 64,
245: 13, 17, 37, 40,
246: 77, 170, 324, 417,
247: 61, 63, 185, 187,
248: 82, 167, 331, 416,
249: 41, 49, 121, 129,
250: 9655, 10039, 19828, 23344,
251: 5, 7, 8, 13,
252: 45, 47, 177, 208,
253: 149, 221, 509, 697,
254: 4, 11, 16, 19,
255: 73, 993, 1559, 1639,
256: 100, 172, 211, 295,

2020-06-19, 17:39   #821
sweety439

Nov 2016

5·571 Posts

Quote:
 Originally Posted by sweety439 the conjectured first 4 Sierpinski/Riesel numbers for bases up to 256
the conjectured first 16 Sierpinski/Riesel numbers for bases up to 149 (will complete to bases up to 2048)
Attached Files
 first 16 Sierpinski CK.txt (12.8 KB, 42 views) first 16 Riesel CK.txt (12.8 KB, 48 views)

2020-06-19, 17:51   #822
sweety439

Nov 2016

5·571 Posts

Quote:
 Originally Posted by sweety439 Stronger conjectures: (assuming k is positive integer) * if k < 78557, then there are infinitely many primes of the form (k*2^n+1)/gcd(k+1,2-1) with integer n>=1 * if k < 11047, then there are infinitely many primes of the form (k*3^n+1)/gcd(k+1,3-1) with integer n>=1 * if k < 419, then there are infinitely many primes of the form (k*4^n+1)/gcd(k+1,4-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*5^n+1)/gcd(k+1,5-1) with integer n>=1 * if k < 174308, then there are infinitely many primes of the form (k*6^n+1)/gcd(k+1,6-1) with integer n>=1 * if k < 209, then there are infinitely many primes of the form (k*7^n+1)/gcd(k+1,7-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*8^n+1)/gcd(k+1,8-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*9^n+1)/gcd(k+1,9-1) with integer n>=1 * if k < 989, then there are infinitely many primes of the form (k*10^n+1)/gcd(k+1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n+1)/gcd(k+1,11-1) with integer n>=1 * if k < 521, then there are infinitely many primes of the form (k*12^n+1)/gcd(k+1,12-1) with integer n>=1 * if k < 15, then there are infinitely many primes of the form (k*13^n+1)/gcd(k+1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n+1)/gcd(k+1,14-1) with integer n>=1 * if k < 673029, then there are infinitely many primes of the form (k*15^n+1)/gcd(k+1,15-1) with integer n>=1 * if k < 38, then there are infinitely many primes of the form (k*16^n+1)/gcd(k+1,16-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*17^n+1)/gcd(k+1,17-1) with integer n>=1 * if k < 398, then there are infinitely many primes of the form (k*18^n+1)/gcd(k+1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n+1)/gcd(k+1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n+1)/gcd(k+1,20-1) with integer n>=1 * if k < 23, then there are infinitely many primes of the form (k*21^n+1)/gcd(k+1,21-1) with integer n>=1 * if k < 2253, then there are infinitely many primes of the form (k*22^n+1)/gcd(k+1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n+1)/gcd(k+1,23-1) with integer n>=1 * if k < 30651, then there are infinitely many primes of the form (k*24^n+1)/gcd(k+1,24-1) with integer n>=1 * if k < 79, then there are infinitely many primes of the form (k*25^n+1)/gcd(k+1,25-1) with integer n>=1 * if k < 221, then there are infinitely many primes of the form (k*26^n+1)/gcd(k+1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n+1)/gcd(k+1,27-1) with integer n>=1 * if k < 4554, then there are infinitely many primes of the form (k*28^n+1)/gcd(k+1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n+1)/gcd(k+1,29-1) with integer n>=1 * if k < 867, then there are infinitely many primes of the form (k*30^n+1)/gcd(k+1,30-1) with integer n>=1 * if k < 239, then there are infinitely many primes of the form (k*31^n+1)/gcd(k+1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n+1)/gcd(k+1,32-1) with integer n>=1 * if k < 511, then there are infinitely many primes of the form (k*33^n+1)/gcd(k+1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n+1)/gcd(k+1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n+1)/gcd(k+1,35-1) with integer n>=1 * if k < 1886, then there are infinitely many primes of the form (k*36^n+1)/gcd(k+1,36-1) with integer n>=1 * if k < 39, then there are infinitely many primes of the form (k*37^n+1)/gcd(k+1,37-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*38^n+1)/gcd(k+1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n+1)/gcd(k+1,39-1) with integer n>=1 * if k < 47723, then there are infinitely many primes of the form (k*40^n+1)/gcd(k+1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n+1)/gcd(k+1,41-1) with integer n>=1 * if k < 13372, then there are infinitely many primes of the form (k*42^n+1)/gcd(k+1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n+1)/gcd(k+1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n+1)/gcd(k+1,44-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*45^n+1)/gcd(k+1,45-1) with integer n>=1 * if k < 881, then there are infinitely many primes of the form (k*46^n+1)/gcd(k+1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n+1)/gcd(k+1,47-1) with integer n>=1 * if k < 1219, then there are infinitely many primes of the form (k*48^n+1)/gcd(k+1,48-1) with integer n>=1 * if k < 31, then there are infinitely many primes of the form (k*49^n+1)/gcd(k+1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n+1)/gcd(k+1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n+1)/gcd(k+1,51-1) with integer n>=1 * if k < 28674, then there are infinitely many primes of the form (k*52^n+1)/gcd(k+1,52-1) with integer n>=1 * if k < 7, then there are infinitely many primes of the form (k*53^n+1)/gcd(k+1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n+1)/gcd(k+1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n+1)/gcd(k+1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n+1)/gcd(k+1,56-1) with integer n>=1 * if k < 47, then there are infinitely many primes of the form (k*57^n+1)/gcd(k+1,57-1) with integer n>=1 * if k < 488, then there are infinitely many primes of the form (k*58^n+1)/gcd(k+1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n+1)/gcd(k+1,59-1) with integer n>=1 * if k < 16957, then there are infinitely many primes of the form (k*60^n+1)/gcd(k+1,60-1) with integer n>=1 * if k < 63, then there are infinitely many primes of the form (k*61^n+1)/gcd(k+1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n+1)/gcd(k+1,62-1) with integer n>=1 * if k < 1589, then there are infinitely many primes of the form (k*63^n+1)/gcd(k+1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n+1)/gcd(k+1,64-1) with integer n>=1 * if k < 509203, then there are infinitely many primes of the form (k*2^n-1)/gcd(k-1,2-1) with integer n>=1 * if k < 12119, then there are infinitely many primes of the form (k*3^n-1)/gcd(k-1,3-1) with integer n>=1 * if k < 361, then there are infinitely many primes of the form (k*4^n-1)/gcd(k-1,4-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*5^n-1)/gcd(k-1,5-1) with integer n>=1 * if k < 84687, then there are infinitely many primes of the form (k*6^n-1)/gcd(k-1,6-1) with integer n>=1 * if k < 457, then there are infinitely many primes of the form (k*7^n-1)/gcd(k-1,7-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*8^n-1)/gcd(k-1,8-1) with integer n>=1 * if k < 41, then there are infinitely many primes of the form (k*9^n-1)/gcd(k-1,9-1) with integer n>=1 * if k < 334, then there are infinitely many primes of the form (k*10^n-1)/gcd(k-1,10-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*11^n-1)/gcd(k-1,11-1) with integer n>=1 * if k < 376, then there are infinitely many primes of the form (k*12^n-1)/gcd(k-1,12-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*13^n-1)/gcd(k-1,13-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*14^n-1)/gcd(k-1,14-1) with integer n>=1 * if k < 622403, then there are infinitely many primes of the form (k*15^n-1)/gcd(k-1,15-1) with integer n>=1 * if k < 100, then there are infinitely many primes of the form (k*16^n-1)/gcd(k-1,16-1) with integer n>=1 * if k < 49, then there are infinitely many primes of the form (k*17^n-1)/gcd(k-1,17-1) with integer n>=1 * if k < 246, then there are infinitely many primes of the form (k*18^n-1)/gcd(k-1,18-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*19^n-1)/gcd(k-1,19-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*20^n-1)/gcd(k-1,20-1) with integer n>=1 * if k < 45, then there are infinitely many primes of the form (k*21^n-1)/gcd(k-1,21-1) with integer n>=1 * if k < 2738, then there are infinitely many primes of the form (k*22^n-1)/gcd(k-1,22-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*23^n-1)/gcd(k-1,23-1) with integer n>=1 * if k < 32336, then there are infinitely many primes of the form (k*24^n-1)/gcd(k-1,24-1) with integer n>=1 * if k < 105, then there are infinitely many primes of the form (k*25^n-1)/gcd(k-1,25-1) with integer n>=1 * if k < 149, then there are infinitely many primes of the form (k*26^n-1)/gcd(k-1,26-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*27^n-1)/gcd(k-1,27-1) with integer n>=1 * if k < 3769, then there are infinitely many primes of the form (k*28^n-1)/gcd(k-1,28-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*29^n-1)/gcd(k-1,29-1) with integer n>=1 * if k < 4928, then there are infinitely many primes of the form (k*30^n-1)/gcd(k-1,30-1) with integer n>=1 * if k < 145, then there are infinitely many primes of the form (k*31^n-1)/gcd(k-1,31-1) with integer n>=1 * if k < 10, then there are infinitely many primes of the form (k*32^n-1)/gcd(k-1,32-1) with integer n>=1 * if k < 545, then there are infinitely many primes of the form (k*33^n-1)/gcd(k-1,33-1) with integer n>=1 * if k < 6, then there are infinitely many primes of the form (k*34^n-1)/gcd(k-1,34-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*35^n-1)/gcd(k-1,35-1) with integer n>=1 * if k < 33791, then there are infinitely many primes of the form (k*36^n-1)/gcd(k-1,36-1) with integer n>=1 * if k < 29, then there are infinitely many primes of the form (k*37^n-1)/gcd(k-1,37-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*38^n-1)/gcd(k-1,38-1) with integer n>=1 * if k < 9, then there are infinitely many primes of the form (k*39^n-1)/gcd(k-1,39-1) with integer n>=1 * if k < 25462, then there are infinitely many primes of the form (k*40^n-1)/gcd(k-1,40-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*41^n-1)/gcd(k-1,41-1) with integer n>=1 * if k < 15137, then there are infinitely many primes of the form (k*42^n-1)/gcd(k-1,42-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*43^n-1)/gcd(k-1,43-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*44^n-1)/gcd(k-1,44-1) with integer n>=1 * if k < 93, then there are infinitely many primes of the form (k*45^n-1)/gcd(k-1,45-1) with integer n>=1 * if k < 928, then there are infinitely many primes of the form (k*46^n-1)/gcd(k-1,46-1) with integer n>=1 * if k < 5, then there are infinitely many primes of the form (k*47^n-1)/gcd(k-1,47-1) with integer n>=1 * if k < 3226, then there are infinitely many primes of the form (k*48^n-1)/gcd(k-1,48-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*49^n-1)/gcd(k-1,49-1) with integer n>=1 * if k < 16, then there are infinitely many primes of the form (k*50^n-1)/gcd(k-1,50-1) with integer n>=1 * if k < 25, then there are infinitely many primes of the form (k*51^n-1)/gcd(k-1,51-1) with integer n>=1 * if k < 25015, then there are infinitely many primes of the form (k*52^n-1)/gcd(k-1,52-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*53^n-1)/gcd(k-1,53-1) with integer n>=1 * if k < 21, then there are infinitely many primes of the form (k*54^n-1)/gcd(k-1,54-1) with integer n>=1 * if k < 13, then there are infinitely many primes of the form (k*55^n-1)/gcd(k-1,55-1) with integer n>=1 * if k < 20, then there are infinitely many primes of the form (k*56^n-1)/gcd(k-1,56-1) with integer n>=1 * if k < 144, then there are infinitely many primes of the form (k*57^n-1)/gcd(k-1,57-1) with integer n>=1 * if k < 547, then there are infinitely many primes of the form (k*58^n-1)/gcd(k-1,58-1) with integer n>=1 * if k < 4, then there are infinitely many primes of the form (k*59^n-1)/gcd(k-1,59-1) with integer n>=1 * if k < 20558, then there are infinitely many primes of the form (k*60^n-1)/gcd(k-1,60-1) with integer n>=1 * if k < 125, then there are infinitely many primes of the form (k*61^n-1)/gcd(k-1,61-1) with integer n>=1 * if k < 8, then there are infinitely many primes of the form (k*62^n-1)/gcd(k-1,62-1) with integer n>=1 * if k < 857, then there are infinitely many primes of the form (k*63^n-1)/gcd(k-1,63-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*64^n-1)/gcd(k-1,64-1) with integer n>=1
Also, (for larger power-of-2 bases)

* if k < 44, then there are infinitely many primes of the form (k*128^n+1)/gcd(k+1,128-1) with integer n>=1
* if k < 38, then there are infinitely many primes of the form (k*256^n+1)/gcd(k+1,256-1) with integer n>=1
* if k < 18, then there are infinitely many primes of the form (k*512^n+1)/gcd(k+1,512-1) with integer n>=1
* if k < 81, then there are infinitely many primes of the form (k*1024^n+1)/gcd(k+1,1024-1) with integer n>=1

* if k < 44, then there are infinitely many primes of the form (k*128^n-1)/gcd(k-1,128-1) with integer n>=1
* if k < 100, then there are infinitely many primes of the form (k*256^n-1)/gcd(k-1,256-1) with integer n>=1
* if k < 14, then there are infinitely many primes of the form (k*512^n-1)/gcd(k-1,512-1) with integer n>=1
* if k < 81, then there are infinitely many primes of the form (k*1024^n-1)/gcd(k-1,1024-1) with integer n>=1

2020-06-19, 17:56   #823
sweety439

Nov 2016

5×571 Posts

Quote:
 Originally Posted by sweety439 Also, (for larger power-of-2 bases) * if k < 44, then there are infinitely many primes of the form (k*128^n+1)/gcd(k+1,128-1) with integer n>=1 * if k < 38, then there are infinitely many primes of the form (k*256^n+1)/gcd(k+1,256-1) with integer n>=1 * if k < 18, then there are infinitely many primes of the form (k*512^n+1)/gcd(k+1,512-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*1024^n+1)/gcd(k+1,1024-1) with integer n>=1 * if k < 44, then there are infinitely many primes of the form (k*128^n-1)/gcd(k-1,128-1) with integer n>=1 * if k < 100, then there are infinitely many primes of the form (k*256^n-1)/gcd(k-1,256-1) with integer n>=1 * if k < 14, then there are infinitely many primes of the form (k*512^n-1)/gcd(k-1,512-1) with integer n>=1 * if k < 81, then there are infinitely many primes of the form (k*1024^n-1)/gcd(k-1,1024-1) with integer n>=1
Some k's have algebra factors, so there are additional conditions for some conjectures:

* For (k*128^n+1)/gcd(k+1,128-1), k is not seventh power of integer nor of the form 2^r with integer r == 3 or 5 or 6 mod 7
* For (k*256^n+1)/gcd(k+1,256-1), k is not of the form 4*q^4 with integer q
* For (k*512^n+1)/gcd(k+1,512-1), k is not cube of integer
* For (k*1024^n+1)/gcd(k+1,1024-1), k is not fifth power of integer

* For (k*128^n-1)/gcd(k-1,128-1), k is not seventh power of integer
* For (k*256^n-1)/gcd(k-1,256-1), k is not square of integer
* For (k*512^n-1)/gcd(k-1,512-1), k is not cube of integer
* For (k*1024^n-1)/gcd(k-1,1024-1), k is not square of integer nor fifth power of integer

2020-06-19, 18:00   #824
sweety439

Nov 2016

5×571 Posts

Quote:
 Originally Posted by sweety439 We can also make much stronger conjectures (the 1st, 2nd, 3rd, and 4th Sierpinski/Riesel conjectures): If k < 4th CK and does not equal to 1st CK, 2nd CK, or 3rd CK, then there are infinitely many primes of the form (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) with integer n>=1 Sierpinski: Code: b: 1st CK, 2nd CK, 3rd CK, 4th CK 2: 78557, 157114, 271129, 271577, 3: 11047, 23789, 27221, 32549, 4: 419, 659, 794, 1466, 5: 7, 11, 31, 35, 6: 174308, 188299, 243417, 282001, 7: 209, 1463, 3305, 3533, 8: 47, 79, 83, 181, 9: 31, 39, 111, 119, 10: 989, 1121, 3653, 3662, 11: 5, 7, 17, 19, 12: 521, 597, 1143, 1509, 13: 15, 27, 47, 71, 14: 4, 11, 19, 26, 15: 673029, 2105431, 2692337, 4621459, 16: 38, 194, 524, 608, 17: 31, 47, 127, 143, 18: 398, 512, 571, 989, 19: 9, 11, 29, 31, 20: 8, 13, 29, 34, 21: 23, 43, 47, 111, 22: 2253, 4946, 6694, 8417, 23: 5, 7, 17, 19, 24: 30651, 66356, 77554, 84766, 25: 79, 103, 185, 287, 26: 221, 284, 1627, 1766, 27: 13, 15, 41, 43, 28: 4554, 8293, 13687, 18996, 29: 4, 7, 11, 19, 30: 867, 9859, 10386, 10570, 31: 239, 293, 521, 1025, 32: 10, 23, 43, 56, 33: 511, 543, 1599, 1631, 34: 6, 29, 41, 64, 35: 5, 7, 17, 19, 36: 1886, 11093, 67896, 123189, 37: 39, 75, 87, 191, 38: 14, 16, 25, 53, 39: 9, 11, 29, 31, 40: 47723, 67241, 68963, 133538, 41: 8, 13, 15, 23, 42: 13372, 30359, 47301, 60758, 43: 21, 23, 65, 67, 44: 4, 11, 19, 26, 45: 47, 91, 231, 275, 46: 881, 1592, 2519, 3104, 47: 5, 7, 8, 16, 48: 1219, 3403, 5531, 5613, 49: 31, 79, 179, 191, 50: 16, 35, 67, 86, 51: 25, 27, 77, 79, 52: 28674, 57398, 83262, 117396, 53: 7, 11, 31, 35, 54: 21, 34, 76, 89, 55: 13, 15, 41, 43, 56: 20, 37, 77, 94, 57: 47, 175, 231, 311, 58: 488, 1592, 7766, 8312, 59: 4, 5, 7, 9, 60: 16957, 84486, 138776, 199103, 61: 63, 123, 311, 371, 62: 8, 13, 29, 34, 63: 1589, 2381, 4827, 7083, 64: 14, 51, 79, 116, 65: 10, 23, 43, 56, 66: 67: 26, 33, 35, 101, 68: 22, 36, 47, 56, 69: 6, 15, 19, 27, 70: 11077, 20591, 22719, 25914, 71: 5, 7, 17, 19, 72: 731, 1313, 1461, 3724, 73: 47, 223, 255, 295, 74: 4, 11, 19, 26, 75: 37, 39, 113, 115, 76: 34, 43, 111, 120, 77: 7, 11, 14, 25, 78: 96144, 186123, 288507, 390656, 79: 9, 11, 29, 31, 80: 1039, 3181, 7438, 12211, 81: 575, 649, 655, 1167, 82: 19587, 29051, 37847, 46149, 83: 5, 7, 8, 13, 84: 16, 69, 101, 154, 85: 87, 171, 431, 515, 86: 28, 59, 115, 146, 87: 21, 23, 65, 67, 88: 26, 179, 311, 521, 89: 4, 11, 19, 23, 90: 27, 64, 118, 155, 91: 45, 47, 137, 139, 92: 32, 61, 125, 154, 93: 95, 187, 471, 563, 94: 39, 56, 134, 151, 95: 5, 7, 17, 19, 96: 68869, 353081, 426217, 427383, 97: 127, 223, 575, 671, 98: 10, 16, 23, 38, 99: 9, 11, 29, 31, 100: 62, 233, 332, 836, 101: 7, 11, 16, 31, 102: 293, 1342, 6060, 6240, 103: 25, 27, 77, 79, 104: 4, 6, 8, 11, 105: 319, 423, 1167, 1271, 106: 2387, 5480, 14819, 17207, 107: 5, 7, 17, 19, 108: 26270, 102677, 131564, 132872, 109: 19, 21, 23, 31, 110: 38, 73, 149, 184, 111: 13, 15, 41, 43, 112: 2261, 2939, 3502, 5988, 113: 20, 31, 37, 47, 114: 24, 91, 139, 206, 115: 57, 59, 173, 175, 116: 14, 25, 53, 64, 117: 119, 235, 327, 591, 118: 50, 69, 169, 188, 119: 4, 5, 7, 9, 120: 121: 27, 103, 110, 293, 122: 40, 47, 79, 83, 123: 55, 61, 63, 69, 124: 31001, 56531, 77381, 145994, 125: 7, 8, 11, 13, 126: 766700, 1835532, 2781934, 2986533, 127: 6343, 7909, 12923, 13701, 128: 44, 85, 98, 173, 129: 14, 51, 79, 116, 130: 1049, 2432, 7073, 9602, 131: 5, 7, 10, 17, 132: 13, 20, 113, 153, 133: 59, 135, 267, 671, 134: 4, 11, 19, 26, 135: 33, 35, 101, 103, 136: 29180, 90693, 151660, 243037, 137: 22, 23, 31, 47, 138: 2781, 3752, 4308, 7229, 139: 6, 9, 11, 13, 140: 46, 95, 187, 236, 141: 143, 283, 711, 851, 142: 12, 131, 155, 221, 143: 5, 7, 17, 19, 144: 59, 86, 204, 231, 145: 1023, 1167, 2159, 2367, 146: 8, 13, 29, 34, 147: 73, 75, 221, 223, 148: 3128, 4022, 4471, 7749, 149: 4, 7, 11, 19, 150: 49074, 95733, 539673, 611098, 151: 37, 39, 113, 115, 152: 16, 35, 67, 86, 153: 15, 34, 43, 55, 154: 61, 94, 216, 249, 155: 5, 7, 14, 17, 156: 157: 47, 59, 159, 191, 158: 52, 107, 122, 211, 159: 9, 11, 29, 31, 160: 22, 139, 183, 300, 161: 95, 127, 287, 319, 162: 6193, 6682, 7336, 14343, 163: 81, 83, 245, 247, 164: 4, 10, 11, 19, 165: 167, 331, 831, 995, 166: 335, 5510, 7349, 9854, 167: 5, 7, 8, 13, 168: 9244, 9658, 15638, 20357, 169: 16, 31, 39, 69, 170: 20, 37, 77, 94, 171: 85, 87, 257, 259, 172: 62, 108, 836, 1070, 173: 7, 11, 28, 31, 174: 6, 29, 41, 64, 175: 21, 23, 65, 67, 176: 58, 119, 235, 296, 177: 79, 447, 1247, 1423, 178: 569, 797, 953, 1031, 179: 4, 5, 7, 9, 180: 1679679, 181: 15, 27, 51, 64, 182: 23, 62, 121, 211, 183: 45, 47, 69, 101, 184: 36, 149, 221, 269, 185: 23, 31, 32, 61, 186: 67, 120, 254, 307, 187: 47, 83, 93, 95, 188: 8, 13, 29, 34, 189: 19, 31, 39, 56, 190: 2157728, 3146151, 3713039, 4352889, 191: 5, 7, 17, 19, 192: 7879, 8686, 17371, 19494, 193: 2687, 6015, 6207, 9343, 194: 4, 11, 14, 19, 195: 13, 15, 41, 43, 196: 16457, 78689, 86285, 95147, 197: 7, 10, 11, 23, 198: 4105, 19484, 21649, 23581, 199: 9, 11, 29, 31, 200: 47, 68, 103, 118, 201: 607, 807, 2223, 2423, 202: 57, 146, 260, 349, 203: 5, 7, 16, 17, 204: 81, 124, 286, 329, 205: 207, 411, 1031, 1235, 206: 22, 47, 91, 116, 207: 25, 27, 77, 79, 208: 56, 98, 153, 265, 209: 4, 6, 8, 11, 210: 211: 105, 107, 317, 319, 212: 70, 143, 283, 285, 213: 51, 215, 339, 427, 214: 44, 171, 236, 259, 215: 5, 7, 17, 19, 216: 92, 125, 309, 342, 217: 655, 863, 871, 919, 218: 74, 145, 293, 364, 219: 9, 11, 21, 23, 220: 50, 103, 118, 324, 221: 7, 11, 31, 35, 222: 333163, 352341, 389359, 410098, 223: 13, 15, 41, 43, 224: 4, 11, 19, 26, 225: 3391, 3615, 10623, 10847, 226: 2915, 11744, 12563, 15704, 227: 5, 7, 17, 19, 228: 1146, 7098, 8474, 25647, 229: 19, 24, 31, 47, 230: 8, 10, 13, 23, 231: 57, 59, 173, 175, 232: 2564, 18992, 27527, 46520, 233: 14, 23, 25, 31, 234: 46, 189, 281, 424, 235: 107, 117, 119, 255, 236: 80, 157, 317, 394, 237: 15, 27, 50, 67, 238: 34571, 36746, 42449, 48038, 239: 4, 5, 7, 9, 240: 1722187, 1933783, 2799214, 241: 175, 287, 527, 639, 242: 8, 16, 38, 47, 243: 121, 123, 285, 365, 244: 6, 29, 41, 64, 245: 7, 11, 31, 35, 246: 77, 170, 324, 417, 247: 61, 63, 185, 187, 248: 82, 167, 331, 416, 249: 31, 39, 111, 119, 250: 9788, 23885, 33539, 50450, 251: 5, 7, 8, 13, 252: 45, 116, 144, 208, 253: 255, 327, 507, 691, 254: 4, 11, 16, 19, 255: 245, 365, 493, 499, 256: 38, 194, 467, 524, Riesel: Code: b: 1st CK, 2nd CK, 3rd CK, 4th CK 2: 509203, 762701, 777149, 784109, 3: 12119, 20731, 21997, 28297, 4: 361, 919, 1114, 1444, 5: 13, 17, 37, 41, 6: 84687, 133946, 176602, 213410, 7: 457, 1291, 3199, 3313, 8: 14, 112, 116, 148, 9: 41, 49, 74, 121, 10: 334, 1585, 1882, 3340, 11: 5, 7, 17, 19, 12: 376, 742, 1288, 1364, 13: 29, 41, 69, 85, 14: 4, 11, 19, 26, 15: 622403, 1346041, 2742963, 16: 100, 172, 211, 295, 17: 49, 59, 65, 86, 18: 246, 664, 723, 837, 19: 9, 11, 29, 31, 20: 8, 13, 29, 34, 21: 45, 65, 133, 153, 22: 2738, 4461, 6209, 8902, 23: 5, 7, 17, 19, 24: 32336, 69691, 109054, 124031, 25: 105, 129, 211, 313, 26: 149, 334, 1892, 1987, 27: 13, 15, 41, 43, 28: 3769, 9078, 14472, 18211, 29: 4, 9, 11, 13, 30: 4928, 5331, 7968, 8958, 31: 145, 265, 443, 493, 32: 10, 23, 43, 56, 33: 545, 577, 764, 1633, 34: 6, 29, 41, 64, 35: 5, 7, 17, 19, 36: 33791, 79551, 89398, 116364, 37: 29, 77, 113, 163, 38: 13, 14, 25, 53, 39: 9, 11, 29, 31, 40: 25462, 29437, 38539, 52891, 41: 8, 13, 17, 25, 42: 15137, 28594, 45536, 62523, 43: 21, 23, 65, 67, 44: 4, 11, 19, 26, 45: 93, 137, 277, 321, 46: 928, 3754, 4078, 4636, 47: 5, 7, 13, 14, 48: 3226, 4208, 7029, 7965, 49: 81, 129, 229, 241, 50: 16, 35, 67, 86, 51: 25, 27, 77, 79, 52: 25015, 25969, 35299, 60103, 53: 13, 17, 37, 41, 54: 21, 34, 76, 89, 55: 13, 15, 41, 43, 56: 20, 37, 77, 94, 57: 144, 177, 233, 289, 58: 547, 919, 1408, 1957, 59: 4, 5, 7, 9, 60: 20558, 80885, 135175, 202704, 61: 125, 185, 373, 433, 62: 8, 13, 29, 34, 63: 857, 3113, 5559, 6351, 64: 14, 51, 79, 116, 65: 10, 23, 43, 56, 66: 67: 33, 35, 37, 101, 68: 22, 43, 47, 61, 69: 6, 9, 21, 29, 70: 853, 4048, 6176, 15690, 71: 5, 7, 17, 19, 72: 293, 2481, 3722, 4744, 73: 112, 177, 297, 329, 74: 4, 11, 19, 26, 75: 37, 39, 113, 115, 76: 34, 43, 111, 120, 77: 13, 14, 17, 25, 78: 90059, 192208, 294592, 384571, 79: 9, 11, 29, 31, 80: 253, 1037, 6148, 11765, 81: 74, 575, 657, 737, 82: 22326, 36438, 44572, 64905, 83: 5, 7, 8, 13, 84: 16, 69, 101, 154, 85: 173, 257, 517, 601, 86: 28, 59, 115, 146, 87: 21, 23, 65, 67, 88: 571, 862, 898, 961, 89: 4, 11, 17, 19, 90: 27, 64, 118, 155, 91: 45, 47, 137, 139, 92: 32, 61, 125, 154, 93: 189, 281, 565, 612, 94: 39, 56, 134, 151, 95: 5, 7, 17, 19, 96: 38995, 78086, 343864, 540968, 97: 43, 225, 321, 673, 98: 10, 23, 43, 56, 99: 9, 11, 29, 31, 100: 211, 235, 334, 750, 101: 13, 16, 17, 33, 102: 1635, 1793, 4267, 4447, 103: 25, 27, 77, 79, 104: 4, 6, 8, 11, 105: 297, 425, 529, 1273, 106: 13624, 14926, 16822, 19210, 107: 5, 7, 17, 19, 108: 13406, 26270, 43601, 103835, 109: 9, 21, 34, 45, 110: 38, 73, 149, 184, 111: 13, 15, 41, 43, 112: 1357, 3843, 4406, 5084, 113: 20, 37, 49, 65, 114: 24, 91, 139, 206, 115: 57, 59, 173, 175, 116: 14, 25, 53, 64, 117: 149, 221, 237, 353, 118: 50, 69, 169, 188, 119: 4, 5, 7, 9, 120: 121: 100, 163, 211, 232, 122: 14, 40, 83, 112, 123: 13, 61, 63, 154, 124: 92881, 104716, 124009, 170386, 125: 8, 13, 17, 29, 126: 480821, 2767077, 3925190, 127: 2593, 3251, 3353, 6451, 128: 44, 59, 85, 86, 129: 14, 51, 79, 116, 130: 2563, 5896, 11134, 26632, 131: 5, 7, 10, 17, 132: 20, 69, 113, 153, 133: 17, 233, 269, 273, 134: 4, 11, 19, 26, 135: 33, 35, 101, 103, 136: 22195, 47677, 90693, 151660, 137: 17, 22, 25, 47, 138: 1806, 4727, 5283, 6254, 139: 6, 9, 11, 13, 140: 46, 95, 187, 236, 141: 285, 425, 853, 993, 142: 12, 131, 155, 219, 143: 5, 7, 17, 19, 144: 59, 86, 204, 231, 145: 1169, 1313, 3505, 3649, 146: 8, 13, 29, 34, 147: 73, 75, 221, 223, 148: 1936, 5214, 5663, 6557, 149: 4, 9, 11, 13, 150: 49074, 95733, 228764, 539673, 151: 37, 39, 113, 115, 152: 16, 35, 67, 86, 153: 34, 43, 57, 65, 154: 61, 94, 216, 249, 155: 5, 7, 14, 17, 156: 157: 17, 69, 101, 217, 158: 52, 107, 211, 266, 159: 9, 11, 29, 31, 160: 22, 139, 183, 253, 161: 65, 97, 257, 289, 162: 3259, 4726, 9292, 16299, 163: 81, 83, 245, 247, 164: 4, 10, 11, 19, 165: 79, 333, 497, 646, 166: 4174, 9019, 11023, 15532, 167: 5, 7, 8, 13, 168: 4744, 14676, 15393, 20827, 169: 16, 33, 41, 49, 170: 20, 37, 77, 94, 171: 85, 87, 257, 259, 172: 235, 982, 1108, 1171, 173: 13, 17, 28, 37, 174: 6, 21, 29, 41, 175: 21, 23, 65, 67, 176: 58, 119, 235, 296, 177: 209, 268, 577, 1156, 178: 22, 79, 87, 334, 179: 4, 5, 7, 9, 180: 181: 25, 27, 29, 41, 182: 62, 121, 245, 304, 183: 45, 47, 137, 139, 184: 36, 149, 221, 334, 185: 17, 25, 32, 61, 186: 67, 120, 254, 307, 187: 51, 79, 93, 95, 188: 8, 13, 29, 34, 189: 9, 21, 39, 49, 190: 626861, 2121627, 3182252, 3749140, 191: 5, 7, 17, 19, 192: 13897, 19492, 20459, 22968, 193: 484, 5350, 6209, 6401, 194: 4, 11, 14, 19, 195: 13, 15, 41, 43, 196: 1267, 16654, 17920, 20692, 197: 10, 13, 17, 23, 198: 3662, 8425, 10546, 13224, 199: 9, 11, 29, 31, 200: 68, 133, 268, 269, 201: 809, 1009, 2425, 2625, 202: 57, 146, 260, 349, 203: 5, 7, 14, 16, 204: 81, 124, 286, 329, 205: 25, 361, 413, 617, 206: 22, 47, 91, 116, 207: 25, 27, 77, 79, 208: 56, 153, 186, 265, 209: 4, 6, 8, 11, 210: 211: 100, 105, 107, 317, 212: 70, 143, 149, 179, 213: 57, 73, 181, 429, 214: 44, 171, 259, 386, 215: 5, 7, 17, 19, 216: 92, 125, 309, 342, 217: 337, 353, 409, 441, 218: 74, 145, 293, 364, 219: 9, 11, 21, 23, 220: 103, 118, 324, 339, 221: 13, 17, 37, 38, 222: 88530, 90091, 282094, 514016, 223: 13, 15, 41, 43, 224: 4, 11, 19, 26, 225: 3617, 3841, 10849, 11073, 226: 820, 12790, 50257, 53398, 227: 5, 7, 17, 19, 228: 16718, 33891, 35267, 41219, 229: 9, 21, 24, 49, 230: 8, 10, 13, 23, 231: 57, 59, 173, 175, 232: 27760, 72817, 98791, 100576, 233: 14, 17, 25, 53, 234: 46, 189, 281, 424, 235: 64, 117, 119, 172, 236: 80, 157, 317, 394, 237: 29, 33, 41, 50, 238: 17926, 34810, 93628, 99094, 239: 4, 5, 7, 9, 240: 2952972, 2985025, 3695736, 4812046, 241: 65, 177, 417, 529, 242: 14, 73, 101, 116, 243: 121, 123, 365, 367, 244: 6, 29, 41, 64, 245: 13, 17, 37, 40, 246: 77, 170, 324, 417, 247: 61, 63, 185, 187, 248: 82, 167, 331, 416, 249: 41, 49, 121, 129, 250: 9655, 10039, 19828, 23344, 251: 5, 7, 8, 13, 252: 45, 47, 177, 208, 253: 149, 221, 509, 697, 254: 4, 11, 16, 19, 255: 73, 993, 1559, 1639, 256: 100, 172, 211, 295,
Corrected: the 4th CK of R2 is 790841, not 784109, this error is because I only searched the primes <= 50000 and only searched (k*b^n+-1)/gcd(k+-1,b-1) for n<=2000

 2020-06-19, 18:06 #825 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 226468 Posts what is your reason for quoting huge blocks of text that you posted on the same day?

 Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 sweety439 14 2021-02-15 15:58 sweety439 sweety439 11 2020-09-23 01:42 sweety439 sweety439 20 2020-07-03 17:22 robert44444uk Conjectures 'R Us 139 2007-12-17 05:17 rogue Conjectures 'R Us 11 2007-12-17 05:08

All times are UTC. The time now is 01:40.

Mon Jun 14 01:40:47 UTC 2021 up 16 days, 23:28, 0 users, load averages: 1.66, 1.65, 1.73