The extensibility of Diophantine pairs [formula omitted

Abstract

A set of m distinct positive integers { a 1 , … , a m } is called a Diophantine m-tuple if a i a j + 1 is a square for each 1 ⩽ i < j ⩽ m . In this paper, we show that for each integer k ⩾ 2 the Diophantine pair { k − 1 , k + 1 } cannot be extended to a Diophantine quintuple.