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Old 2017-05-02, 11:14   #8
Nov 2016

281910 Posts

These problems are to find a prime of the form (k*b^n+c)/gcd(k+c,b-1) with integer n>=1 for fixed integers k, b and c, k>=1, b>=2, gcd(k,c)=1 and gcd(b,c)=1.

For some (k,b,c), there cannot be any prime because of covering set (e.g. (k,b,c) = (78557,2,1), (334,10,-1) or (84687,6,-1)) or full algebra factors (e.g. (k,b,c) = (9,4,-1), (2500,16,1) or (9,4,-25) (the case (9,4,-25) can produce prime only for n=1)) or partial algebra factors (e.g. (k,b,c) = (144,28,-1), (25,17,-9) or (1369,30,-1)). It is conjectured that for every (k,b,c) which cannot be proven that they do not have any prime, there are infinitely primes of the form (k*b^n+c)/gcd(k+c,b-1). (Notice the special case: (k,b,c) = (8,128,1), it cannot have any prime but have neither covering set nor algebra factors)

However, there are many such cases even not have a single known prime, like (21181,2,1), (2293,2,-1), (4,53,1), (1,185,-1), (1,38,1), (269,10,1), (197,7,-1), (4105,17,-9), (16,21,335), (5,36,821), but not all case will produce a minimal prime to base b, e.g. the form (197*7^n-1)/2 is the form 200{3} in base 7, but since 2 is already prime, the smallest prime of this form (if exists) will not be a minimal prime in base 7.

The c=1 and gcd(k+c,b-1)=1 case is the Sierpinski problem base b, and the c=-1 and gcd(k+c,b-1)=1 case is the Riesel problem base b.

Last fiddled with by sweety439 on 2017-05-02 at 11:27
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