Reduced diophantine quadruples with the binary recurrence Gn = AGn–1 – Gn–2

Abstract

Given a positive integer A ≠ 2. In this paper, we show that there do not exist two positive integer pairs {a,b} ≠ {c,d} such that the values of ac+1, ad+1 and bc+1, bd+1 are the terms of the sequence {Gn}n≥0 which satisfies the recurrence relation Gn = AGn-1 - Gn-2 with the initial values G0 = 0, G1 = 1.

Keywords

• reduced diophantine quadruples
• binary recurrences