# Sigma k(F m)=F n

### New Zealand Journal of Mathematics

Vol. 40, (2010), Pages 1-13

Florian Luca

Mathematical Institute, UNAM

Ap. Postal 61-3 (Xangari), CP 58089

Morelia, Michoacan,

Mexico

&

School of Mathematics

University of the Witwatersrand

P.O. Wits 2050

South Africa

Benne de Weger

Faculty of Mathematics and Computer Science

Eindhoven University of Technology

PO Box 513, 5600 MB Eindhoven

The Netherlands

Abstract Let σk(n) be the sum of the kth powers of the divisors of n. Here, we prove that if $(F_n)_{n\ge 1}$ is the Fibonacci sequence, then the only solutions of the equation σk(Fm) = Fn in positive integers $k\ge 2,~m$ and n have k = 2 and $m\in \{1,2,3\}$. The proof uses linear forms in two and three logarithms, lattice basis reduction, and some elementary considerations.

Keywords Fibonacci numbers, Arithmetic functions

Classification (MSC2000) 11B39