Sigma k(F m)=F n

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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

mailto:fluca@matmor.unam.mx

Benne de Weger

Faculty of Mathematics and Computer Science

Eindhoven University of Technology

PO Box 513, 5600 MB Eindhoven

The Netherlands

mailto:b.m.m.d.weger@tue.nl]




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

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