On Fibonacci numbers with few prime divisors
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 81, Number 2
(2005), 17-20.
Abstract
If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci
number, and $\omega(n)$ for the number of distinct prime divisors
of $n$. We give a description of Fibonacci numbers satisfying
$\omega(F_n) \leq 2$. Moreover, we prove that the inequality
$\omega(F_n) \geq (\log n)^{\log 2 + o(1)}$ holds for almost
all $n$. We conjecture that $\omega(F_n) \gg \log n$ for composite
$n$, and give a heuristic argument in support of this conjecture.
Secondary Subjects:
11K65
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.pja/1116442053
Mathematical Reviews number (MathSciNet): MR2126070
Zentralblatt MATH identifier: 1087.11009
Digital Object Identifier: doi:10.3792/pjaa.81.17
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