ON THE FAMILY OF DIOPHANTINE TRIPLES { k − 1, k + 1, 16 k 3 − 4 k }

Source abstract

Abstract It is proven that if k ≥ 2 is an integer and 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 = 4 k or d = 64 k 5 −48 k 3 +8 k . Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form { k − 1, k + 1, c , d } are regular.