There is no Diophantine D(−1)$D(-1)$‐quadruple

A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine D(−1)$D(-1)$‐set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine D(−1)$D(-1)$‐quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical D(−1)$D(-1)$‐quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine D(−1)$D(-1)$‐quadruples.