Complete Solution of Diophantine Pairs Induced by Some Fibonacci Formula.

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]