In this paper, applying the Primitive Divisor Theorem, we solve completely the diophantine equation
in non-negative integers , and if the binary recurrence is either the Fibonacci sequence, or the Lucas sequence, or it satisfies the recurrence relation with , and an arbitrary positive integer . The more specific case
was investigated by Marques  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. 213–220DOI 10.4171/PM/1914