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

Abstract

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 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 = 3. And for k = 3 we prove that if the set {1, 10, 13, d} is a D(−9)-quadruple, then d = 45.