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

TL;DR

This paper presents a precise method leveraging elliptic curves and Heegner points to determine if a number is congruent, enhancing verification accuracy in number theory.

Contribution

It introduces a novel approach using elliptic curve invariants and Heegner points for exact verification of congruent numbers.

Findings

Established the relationship between congruent numbers and elliptic curves.

Computed the conductor of the associated elliptic curve.

Proved that the $L$-series coefficient $a_m$ equals zero when $m ot ot ext{ } ext{mod } 4$.

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 $m \equiv 3 \mod 4$.By using the invariants of the elliptic curve introduced above, we calculate Heegner points to quickly verify whether $n$ is a congruent number.