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

摘要

In this paper, applying the Primitive Divisor Theorem, we solve completely the diophantine equation G_(n1), G_(n2) . . . G_(nk) + 1 = G_m~2. 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+l) . . . G_(n+k-1) + 1 = G_m~2 was investigated by Marques [3] in Portugaliae Mathematica in the case of the Fibonacci sequence.