Proceedings of the Japan Academy, Series A, Mathematical Sciences

On Fibonacci numbers with few prime divisors

Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek

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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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 2 (2005), 17-20.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116442053

Digital Object Identifier
doi:10.3792/pjaa.81.17

Mathematical Reviews number (MathSciNet)
MR2126070

Zentralblatt MATH identifier
1087.11009

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Secondary: 11K65: Arithmetic functions [See also 11Nxx]

Keywords
Fibonacci numbers arithmetic functions prime divisors

Citation

Bugeaud, Yann; Luca, Florian; Mignotte, Maurice; Siksek, Samir. On Fibonacci numbers with few prime divisors. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 2, 17--20. doi:10.3792/pjaa.81.17. https://projecteuclid.org/euclid.pja/1116442053


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