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.
@article{RSMUP_2022__148__23_0,
author = {Reinholz, Lindsey and Yang, Qiduan},
title = {On the extension of even families of non-congruent numbers},
journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
pages = {23--49},
volume = {148},
year = {2022},
doi = {10.4171/rsmup/105},
zbl = {07673820},
mrnumber = {4542371},
language = {en},
url = {http://archive.numdam.org/articles/10.4171/rsmup/105/}
}
TY - JOUR AU - Reinholz, Lindsey AU - Yang, Qiduan TI - On the extension of even families of non-congruent numbers JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2022 SP - 23 EP - 49 VL - 148 UR - http://archive.numdam.org/articles/10.4171/rsmup/105/ DO - 10.4171/rsmup/105 LA - en ID - RSMUP_2022__148__23_0 ER -
%0 Journal Article %A Reinholz, Lindsey %A Yang, Qiduan %T On the extension of even families of non-congruent numbers %J Rendiconti del Seminario Matematico della Università di Padova %D 2022 %P 23-49 %V 148 %U http://archive.numdam.org/articles/10.4171/rsmup/105/ %R 10.4171/rsmup/105 %G en %F RSMUP_2022__148__23_0
Reinholz, Lindsey; Yang, Qiduan. On the extension of even families of non-congruent numbers. Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 23-49. doi : 10.4171/rsmup/105. http://archive.numdam.org/articles/10.4171/rsmup/105/
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