Families of non-$\theta $-congruent numbers with arbitrarily many prime factors
Volume 152 / 2018
Colloquium Mathematicum 152 (2018), 255-271
MSC: Primary 11G05; Secondary 14H52.
DOI: 10.4064/cm7151-4-2017
Published online: 1 March 2018
Abstract
The concept of $\theta $-congruent numbers was introduced by Fujiwara (1998), who showed that for primes $p\equiv 5,7,19 \ ({\rm mod} 24)$, $p$ is not a $\pi /3$-congruent number. We show the existence of two infinite families of composite non-$\pi /3$-congruent numbers and non-$2\pi /3$-congruent numbers, obtained from products of primes which are congruent to $5$ modulo $24$ and to $13$ modulo $24$ respectively. This is achieved by generalizing a result obtained by Serf (1991) based on descent on certain elliptic curves, and by extending a method of Iskra (1996) involving the classical (or $\pi /2$-) congruent numbers.
