Geodesic
Asymptotically exact heuristics for prime divisors of the sequence \(\{a^k+b^k\}^\infty_{k=1}\)
Journal of integer sequences, Tome 9 (2006) no. 2
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.
Classification : 11N37, 11N69, 11R45
Keywords: primitive root, chebotarev density theorem, Dirichlet density 
@article{JIS_2006__9_2_a0,
     author = {Moree,  Pieter},
     title = {Asymptotically exact heuristics for prime divisors of the sequence \(\{a^k+b^k\}^\infty_{k=1}\)},
     journal = {Journal of integer sequences},
     year = {2006},
     volume = {9},
     number = {2},
     zbl = {1107.11038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2006__9_2_a0/}
}
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Moree,  Pieter. Asymptotically exact heuristics for prime divisors of the sequence \(\{a^k+b^k\}^\infty_{k=1}\). Journal of integer sequences, Tome 9 (2006) no. 2. http://geodesic.mathdoc.fr/item/JIS_2006__9_2_a0/