Asymptotically exact heuristics for prime divisors of a^k+b^k
Number Theory
2007-05-23 v1
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.
Cite
@article{arxiv.math/0311483,
title = {Asymptotically exact heuristics for prime divisors of a^k+b^k},
author = {Pieter Moree},
journal= {arXiv preprint arXiv:math/0311483},
year = {2007}
}
Comments
11 pages, 1 table