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Asymptotically exact heuristics for prime divisors of a^k+b^k

Pieter Moree
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Abstract

Let N_{a,b}(x) count the number of primes p<=x with p dividing a^k+b^k for some k>=1. It is known that asymptotically N_{a,b}(x) grows like c(a,b)x/log x for some rational number c(a,b) that depends in a rather intricate way on a and b. A simple heuristic formula for N_{a,b}(x) is proposed and it is proved that it is asymptotically exact, i.e. has the same asymptotic behaviour as N_{a,b}(x). Connections with Ramanujan sums and character sums are discussed.

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Pieter Moree. 2003-11-26. Asymptotically exact heuristics for prime divisors of a^k+b^k. https://arxiv.org/abs/math/0311483

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