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Old 2020-10-31, 00:22   #1076
sweety439
 
Nov 2016

22·691 Posts
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These problems generalized the Sierpinski problem and the Riesel problem to other bases (instead of only base 2), since for bases b>2, k*b^n+1 is always divisible by gcd(k+1,b-1) and k*b^n-1 is always divisible by gcd(k-1,b-1), the formulas are (k*b^n+1)/gcd(k+1,b-1) for Sierpinski and (k*b^n-1)/gcd(k-1,b-1) for Riesel, for a given base b>=2, we will find and proof the smallest k>=1 such that (k*b^n+1)/gcd(k+1,b-1) (for Sierpinski) or (k*b^n-1)/gcd(k-1,b-1) (for Riesel) is not prime for all n>=1, any k that obtains a full covering set in any manner from ALGEBRAIC factors will be excluded, in many instances, this includes k's where there is a partial covering set of numeric factors (or a single numeric factor) and a partial covering set of algebraic factors that combine to make a full covering set, all k below the lowest k found to have a NUMERIC covering set must have a prime including multiples of the base (MOB) but excluding the k's that obtains a full covering set in any manner from ALGEBRAIC factors, for the lowest k found to have a NUMERIC covering set for all bases b<=2048 and b = 4096, 8192, 16384, 32768, 65536, see Sierpinski and Riesel
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