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### 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.

First Page:
Primary Subjects: 11B39
Secondary Subjects: 11K65
Full-text: Open access

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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