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 {k − 1, k + 1, 16k − 4k, d} increased by 1 is a perfect square, then d = 4k or d = 64k−48k+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.
