The congruent numbers have positive natural density

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.