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Old 2020-03-27, 15:45   #29
sweety439's Avatar
Nov 2016

22×3×179 Posts

Originally Posted by R. Gerbicz View Post
gd_barnes is right, to extend his post:
to get a riesel/sierpinski cover you have to use d_i divisors in each remainder class in the covering set, you can't use algebraic factors, you can't use even in those cases where you could use, like in your example for k=81 we could cover the n==0 mod 2 case, because 81*1024^n-1 has an algebraic factor. For this reason we can rule out say k=9, it is sure that this sequence contains no prime, but it has no covering set. The proof could be hard/impossible(?), but just see the prime factorization for n=54: .

ps. you could also cover the n==0 mod 1 so every integer for k=81 with an algebraic factor.
9*1024^n-1 is unlikely to have a covering set, like the simple cases: 3*2^n+-1, 5*2^n+-1, 7*2^n+-1, 9*2^n+-1, 11*2^n+-1, etc. they are unlikely to have a covering set, but also no proof.
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