On the extension of even families of non-congruent numbers
Lindsey Reinholz
University of British Columbia Okanagan, Kelowna, CanadaQiduan Yang
University of British Columbia Okanagan, Kelowna, Canada
![On the extension of even families of non-congruent numbers cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rsmup-volume-148.png&w=3840&q=90)
Abstract
A method that extends existing families of even non-congruent numbers to produce new families of non-congruent numbers with arbitrarily many distinct prime factors is presented. We show that infinitely many new non-congruent numbers can be generated by appending a suitable collection of primes onto any even non-congruent number whose corresponding congruent number elliptic curve has 2-Selmer rank of zero. Our method relies upon Monsky’s formula for computing the 2-Selmer rank of the congruent number elliptic curve. Even non-congruent numbers constructed according to our result have an unlimited number of prime factors in each odd congruence class modulo eight, and have congruent number elliptic curves with 2-Selmer rank equal to zero.
Cite this article
Lindsey Reinholz, Qiduan Yang, On the extension of even families of non-congruent numbers. Rend. Sem. Mat. Univ. Padova 148 (2022), pp. 23–49
DOI 10.4171/RSMUP/105