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

László Szalay

doi:10.4171/pm/1914

JournalspmVol. 69, No. 3pp. 213–220

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

László Szalay
University of West Hungary, Sopron, Hungary

Download PDF

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 $G_{n} = A G_{n - 1} - G_{n - 2}$ 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