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The congruent numbers have positive natural density

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arxiv 1603.08479 v2 pith:2NFLPNNB submitted 2016-03-28 math.NT

Alexander Smith This is my paper · ORCID
classification math.NT
keywords positiveequalintegersleastsquarefreecongruentconjecturenumbers
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We prove that the rational elliptic curve y^2 = x^3 - n^2x satisfies the full Birch and Swinnerton-Dyer conjecture for at least 41.9% of positive squarefree integers n equal to 1, 2, or 3 mod 8, and that it satisfies the regular BSD conjecture for at least 55.9% of positive squarefree integers n equal to 5, 6, or 7 mod 8. In particular, at least 55.9% of positive squarefree integers equal to 5, 6, or 7 mod 8 are congruent numbers. These proofs complete an argument started by Tian, Yuan, and Zhang.

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  1. Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning

    math.NT 2025-09 conditional novelty 4.0 of 10

    A large empirical study of congruent number elliptic curves up to D < 3×10^6 confirms several Selmer-rank heuristics but does not support the paper's headline claim that Goldfeld's conjecture is rigorously verified.

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