Any Diophantine quintuple contains a regular Diophantine quadruple

Abstract

A set { a 1 , … , a m } of m distinct positive integers is called a Diophantine m-tuple if a i a j + 1 is a perfect square for all i, j with 1 ⩽ i < j ⩽ m . It is conjectured that if { a , b , c , d } is a Diophantine quadruple with a < b < c < d , then d = d + , where d + = a + b + c + 2 a b c + 2 r s t and r = a b + 1 , s = a c + 1 , t = b c + 1 . In this paper, we show that if { a , b , c , d , e } is a Diophantine quintuple with a < b < c < d < e , then d = d + .