# Sigma k(F m)=F n

### From NZJM

**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 σ_{k}(*n*) be the sum of the *k*th powers of the divisors of *n*. Here, we prove that if is the Fibonacci sequence, then the only solutions of the equation σ_{k}(*F*_{m}) = *F*_{n} in positive integers and *n* have *k* = 2 and . 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