# Diophantine equations with binary recurrences associated to the Brocard–Ramanujan problem

### László Szalay

University of West Hungary, Sopron, Hungary

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## Abstract

In this paper, applying the Primitive Divisor Theorem, we solve completely the diophantine equation

$G_{n_1}G_{n_2}\dots G_{n_k}+1=G_m^2$

in non-negative integers $k>0$, $m$ and $n_1 < n_2 < \dots < n_k$ if the binary recurrence $\{G_n\}_{n=0}^\infty$ is either the Fibonacci sequence, or the Lucas sequence, or it satisfies the recurrence relation $G_n=AG_{n-1} - G_{n-2}$ with $G_0=0$, $G_1=1$ and an arbitrary positive integer $A$. The more specific case

$G_{n}G_{n+1}\dots G_{n+k-1}+1=G_m^2$

was investigated by Marques [3] in *Portugaliae Mathematica* in the case of the Fibonacci sequence.

## Cite this article

László Szalay, Diophantine equations with binary recurrences associated to the Brocard–Ramanujan problem. Port. Math. 69 (2012), no. 3, pp. 213–220

DOI 10.4171/PM/1914