Complete Solution of Diophantine Pairs Induced by Some Fibonacci Formula

Abstract

A set [Formula: see text] of positive integers is called a Diophantine [Formula: see text]-tuple if [Formula: see text] is a perfect square for all [Formula: see text]. Let [Formula: see text] be the Diophantine triple with [Formula: see text]. In this paper, we find the condition for the reduction of third element [Formula: see text], and using this result, we prove the extendibility of Diophantine pair [Formula: see text], where [Formula: see text] is the [Formula: see text]-th Fibonacci number.