Congruent Numbers and The Rank of Elliptic Curves

摘要

Let p and q be prime with p ≡ 3(mod8) and q ≡ 5(mod8), and let (p/q) = -1. If n = pq, then n is a congruent if and only if the equation 2qw~2 = 1 + p~2z~4 has rational solutions. And The Birch Swinnerton-Dyer conjecture predicts that the rank of E_n(Q) is one.