There is no Diophantine $D(-1)$--quadruple
Number Theory
2022-01-26 v1
Abstract
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 --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 --quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical --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 --quadruple.
Keywords
Cite
@article{arxiv.2010.09200,
title = {There is no Diophantine $D(-1)$--quadruple},
author = {Nicolae Ciprian Bonciocat and Mihai Cipu and Maurice Mignotte},
journal= {arXiv preprint arXiv:2010.09200},
year = {2022}
}
Comments
29 pages, 2 figures