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

First available in Project Euclid: 18 May 2005

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Zentralblatt MATH identifier

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

Fibonacci numbers arithmetic functions prime divisors


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.

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