On the extension of even families of non-congruent numbers

  • Lindsey Reinholz

    University of British Columbia Okanagan, Kelowna, Canada
  • Qiduan Yang

    University of British Columbia Okanagan, Kelowna, Canada
On the extension of even families of non-congruent numbers cover

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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