Open Access
Feb. 2005 On Fibonacci numbers with few prime divisors
Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek
Proc. Japan Acad. Ser. A Math. Sci. 81(2): 17-20 (Feb. 2005). DOI: 10.3792/pjaa.81.17


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.


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Yann Bugeaud. Florian Luca. Maurice Mignotte. Samir Siksek. "On Fibonacci numbers with few prime divisors." Proc. Japan Acad. Ser. A Math. Sci. 81 (2) 17 - 20, Feb. 2005.


Published: Feb. 2005
First available in Project Euclid: 18 May 2005

zbMATH: 1087.11009
MathSciNet: MR2126070
Digital Object Identifier: 10.3792/pjaa.81.17

Primary: 11B39
Secondary: 11K65

Keywords: Arithmetic functions , Fibonacci numbers , prime divisors

Rights: Copyright © 2005 The Japan Academy

Vol.81 • No. 2 • Feb. 2005
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