Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.8

Asymptotically Exact Heuristics for Prime
Divisors of the Sequence $\{a^k+b^k\}_{k=1}^{\infty}$

Pieter Moree
Korteweg-de Vries Instituut
Plantage Muidergracht 24
1018 TV Amsterdam
The Netherlands


Let $N_{a,b}(x)$ count the number of primes $p\le x$ with $p$ dividing $a^k+b^k$ for some $k\ge 1$. It is known that $N_{a,b}(x)\sim c(a,b)x/\log x$ 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 behavior as $N_{a,b}(x)$. Connections with Ramanujan sums and character sums are discussed.

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Received February 14 2005; revised version received February 24 2006. Published in Journal of Integer Sequences July 7 2006.

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