Families of non-congruent numbers with odd prime factors of the form 8k + 3

A congruent number is a positive integer which can be represented as the area of a right triangle such that all of its side lengths are rational numbers. The problem determining whether a given number is congruent is usually studied by computing the Mordell-Weil rank of the corresponding elliptic curve. The Monsky matrix gives a way to compute efficiently the 2-Selmer rank, thereby gives an upper bound for the Mordell-Weil rank. In this paper, by using Monsky's matrix, we present new families of non-congruent numbers such that all of their odd prime factors are of the form 8k+3. Our result generalizes previous works of Reinholz–Spearman–Yang [12] and Cheng–Guo [3].