The extension of the D(−k 2)-pair {k 2,k 2 + 1}

Abstract

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k 2 ,k 2 +1,c,d} is a D(−k 2 )-quadruple with c < d, then c = 1 and d = 4k 2 +1. This extends the work of the first author [20] and that of Dujella [4].