Families of non-congruent numbers with arbitrarily many triplets of prime factors
A natural number \(n\) is a non-congruent number if and only if the Mordell-Weil rank \(r(n)\) of the congruent number elliptic curve given by \(E_n: y^2=x^3-n^2 x\) over the rational numbers is \(0\). Note that \(r(n)\) is bounded by the 2-Selmer rank \(s(n)\) of \(E_n\), so in order to prove that a given number \(n\) is non-congruent, it is enough to show that \(s(n)=0\). Monsky [\textit{D. R. Heath-Brown}, Invent. Math. 118, No. 2, 331--370 (1994; Zbl 0815.11032), Appendix] represented the 2-Selmer rank as the nullity of a square matrix \(M\) over the finite field \(\mathbb{F}_2\), which gives us an explicit formula to compute \(s(n)\).\N\NIn recent years, many authors have constructed families of non-congruent numbers with arbitrarily many prime factors by using Monsky's formula for the 2-Selmer rank. For example, see the references [\textit{L. Reinholz} et al., Int. J. Number Theory 14, No. 3, 669--692 (2018; Zbl 1429.11108); \textit{W. Cheng} and \textit{X. Guo}, J. Number Theory 196, 291--305 (2019; Zbl 1454.11109); \textit{W. Cheng} and \textit{X. Guo}, Int. J. Number Theory 15, No. 4, 677--711 (2019; Zbl 1462.11050)]. Using the same method, this paper shows the existence of infinitely many new families of highly composite non-congruent numbers, obtained as products of arbitrary number of triples of primes of certain patterns (Theorems 1.1--1.3). Finally, the density of such types of non-congruent numbers is discussed.
scientific article; zbMATH DE number 7898739