[QUOTE=sweety439;547944][URL="https://docs.google.com/document/d/e/2PACX-1vTU3PR5EKMbYxPHaSuAmg4TvzoHAcqLsjaSebcpOBYQ1E0bhtzI191OgAwXYXqVcFbCMR31cy0Gs5Te/pub"]Sierpinski problems[/URL]

[URL="https://docs.google.com/document/d/e/2PACX-1vQSfWUMUMXA47_TyBey0pBxMRFnhJN-ExR1JwQMAob208FNXoD-oaAaApuX6cHlMPd3MZS-qU5gDtcD/pub"]Riesel problems[/URL][/QUOTE]

[URL="https://docs.google.com/document/d/e/2PACX-1vTP-T3Qh0JyJiZ8yt2baMNnp4ea9vzbt6orN994Jd7pR9meTSacGMBNCQA_jBULqSwnib6LsEW6UB27/pub"]https://docs.google.com/document/d/e/2PACX-1vTP-T3Qh0JyJiZ8yt2baMNnp4ea9vzbt6orN994Jd7pR9meTSacGMBNCQA_jBULqSwnib6LsEW6UB27/pub[/URL]

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

the conjectured first 4 Sierpinski/Riesel numbers for bases up to 256

[QUOTE=carpetpool;547416](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<n if n is prime --- and there are infinitely many primes, so no finite set is possible.

Back to your original problem, if you "conjecture" there are infinitely many primes of the form (k*b^n+c)/gcd(k+c, b-1), you are really conjecturing that step 3 is true, alongside from conjecturing that if all 4 steps are true, there are infinitely many primes of that form.[/QUOTE]

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 [URL="https://mersenneforum.org/showpost.php?p=547407&postcount=783"]#783[/URL] 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.

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

[QUOTE=sweety439;548484]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[/QUOTE]

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

[QUOTE=sweety439;548484]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[/QUOTE]

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,
[/CODE]

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,
[/CODE]

[QUOTE=sweety439;548276]the conjectured first 4 Sierpinski/Riesel numbers for bases up to 256[/QUOTE]

the conjectured first 16 Sierpinski/Riesel numbers for bases up to 149 (will complete to bases up to 2048)

[QUOTE=sweety439;548484]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[/QUOTE]

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

[QUOTE=sweety439;548517]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[/QUOTE]

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

[QUOTE=sweety439;548512]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,
[/CODE]

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,
[/CODE][/QUOTE]

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

what is your reason for quoting huge blocks of text that you posted on the same day?