Families of non-congruent numbers with odd prime factors of the form 8k + 3

Junguk Lee, Wan Lee, Hayan Nam, Myungjun Yu

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Abstract

A congruent number is a positive integer which can be represented as the area of a right triangle such that all of its side lengths are rational numbers. The problem determining whether a given number is congruent is usually studied by computing the Mordell-Weil rank of the corresponding elliptic curve. The Monsky matrix gives a way to compute efficiently the 2-Selmer rank, thereby gives an upper bound for the Mordell-Weil rank. In this paper, by using Monsky's matrix, we present new families of non-congruent numbers such that all of their odd prime factors are of the form 8k+3. Our result generalizes previous works of Reinholz–Spearman–Yang [12] and Cheng–Guo [3].

Original languageEnglish
JournalJournal of Number Theory
DOIs
Publication statusAccepted/In press - 2021

Bibliographical note

Funding Information:
We are grateful to the anonymous referee for many comments to improve the readability of the manuscript. Junguk Lee was supported by KAIST Advanced Institute for Science-X fellowship. Wan Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C2011017 ). He was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ). Hayan Nam was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A106231911 ). Myungjun Yu was supported by a KIAS Individual Grant ( SP075201 ) via the Center for Mathematical Challenges at KIAS and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01007604 ).

Publisher Copyright:
© 2021 Elsevier Inc.

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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