ON A FAMILY OF DIOPHANTINE TRIPLES {k, A2k+2A, (A+1)2k+2 (A+1)} WITH TWO PARAMETERS

Let A and k be positive integers. We study the Diophantine quadruples {k, A(2)k + 2A, (A + 1)(2)k + 2(A + 1), d}. We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then d = (4A(4) + 8A(3) + 4A(2))k(3) + (16A(3) + 24A(2) + 8A)k(2) + (20A(2) + 20A + 4)k + (8A + 4) when 3 <= A <= 10. This extends a theorem obtained by Dujella [7] for A = 1, and also, a classical theorem of Baker and Davenport [2] for A = k = 1.