The D(-k2)-triple {1,k2+1,k2+4} with k prime

Abstract

Let n be a nonzero integer. A set of m distinct 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 prime number. In this paper we prove that the D(-k^2)-triple {; ; 1, k^2 +1, k^2 +4}; ; cannot be extended to a D(-k 2)-quadruple if k >= 5: And if k = 3 we prove that if the set {; ; 1, 10, 13, d}; ; is a D(-9)-quadruple, then d = 45: