The congruent numbers have positive natural density
Number Theory
2016-03-31 v2
Abstract
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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Cite
@article{arxiv.1603.08479,
title = {The congruent numbers have positive natural density},
author = {Alexander Smith},
journal= {arXiv preprint arXiv:1603.08479},
year = {2016}
}
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32 Pages