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

### László Szalay

University of West Hungary, Sopron, Hungary

## Abstract

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

$G_{n_{1}}G_{n_{2}}…G_{n_{k}}+1=G_{m}$

in non-negative integers $k>0$, $m$ and $n_{1}<n_{2}<⋯<n_{k}$ if the binary recurrence ${G_{n}}_{n=0}$ 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

$G_{n}G_{n+1}…G_{n+k−1}+1=G_{m}$

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