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

Abstract

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

Gn1Gn2Gnk+1=Gm2G_{n_1}G_{n_2}\dots G_{n_k}+1=G_m^2

in non-negative integers k>0k>0, mm and n1<n2<<nkn_1 < n_2 < \dots < n_k if the binary recurrence {Gn}n=0\{G_n\}_{n=0}^\infty is either the Fibonacci sequence, or the Lucas sequence, or it satisfies the recurrence relation Gn=AGn1Gn2G_n=AG_{n-1} - G_{n-2} with G0=0G_0=0, G1=1G_1=1 and an arbitrary positive integer AA. The more specific case

GnGn+1Gn+k1+1=Gm2G_{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