Adaptation of Monsky matrices for 𝜃-congruent numbers

Abstract

[Formula: see text]-congruent numbers are a generalization of congruent numbers. As in the classical case, there is a particular interest in finding infinite families of non-[Formula: see text]-congruent numbers with special properties, such as having an arbitrary number of prime factors. This paper presents methods to find families of non-[Formula: see text]-congruent numbers in the spirit of the initial result of Iskra, and extending the results of Girard, Lalín and Nair for [Formula: see text] and [Formula: see text] by adapting Monsky matrices following the ideas of Reinholz, Spearman and Yang. This paper also presents induction theorems using adapted Monsky matrices in order to find more general non-[Formula: see text]-congruent families than those presently known.