Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Abel Medina Lourenço [a]

Powers of Fibonacci numbers which are products of repdigits

Pages: 205-215
Accepted accepted in revised form: 29 June 2023
Mathematics Subject Classification: 11D61, 11B39.
Keywords: Exponential Diophantine equations, Fibonacci numbers.
[a]: Universidade de São Paulo, Instituto de Matemática e Estatística, São Paulo, Brasil

Abstract: In this article we solve the equation $$F_{n}^{k} = \left(d_{1}\cdot\frac{{10}^{m} -1}{9}\right) \cdot \left(d_{2}\cdot\frac{{10}^{q} -1}{9}\right)$$, with $$n,k,d_{1},d_{2},m,q \in \mathbb{N}, d_{1},d_{2} = 1,\dots,9, m,q \ge 2$$, $$k \ge 2$$, showing that the only perfect power of a Fibonacci number which is a product of two repdigits is $$F_{10}^{2} = 55 \cdot 55$$.
In order to do this we use only elementary methods, like divisibility properties of Fibonacci numbers, periodicity, results on prime factorizations and an application of Nagell-Ljunggren equations.

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