PRIMES IN A PRESCRIBED ARITHMETIC PROGRESSION DIVIDING THE SEQUENCE {a~k + b~k}_k~∞=1

摘要

Given positive integers a, b, c and d such that c and d are coprime, we show that the primes p = c (mod d) dividing a~k + b~k for some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.