An exact way to verify whether n is a congruent number using Heegner points

Abstract

We introduce the relationship between congruent numbers and elliptic curves, and compute the conductor of the elliptic curve $y^2=x^3-n^{2}x$ associated with it. Furthermore, we prove that its $L$-series coefficient $a_m=0$ when $m\equiv 3\mathrm{mod}4$.By using the invariants of the elliptic curve introduced above, we calculate Heegner points to quickly verify whether $n$ is a congruent number.

10 pages,0 figures