Congruent number problem and elliptic curves

Abstract

Let $n$ be a positive integer congruent to 5 modulo 8. Assume that all prime factors of $n$ are congruent to 1 modulo 4 and that $\\BQ(\\sqrt{-n})$ has no ideal class of order 4. In this article, we prove that $n$ is a congruent number, i.e., it is the area of a right triangle with rational side-lengths (see Theorem thm6.1 ). Moreover, we also survey results on BSD conjecture and Goldfeld conjecture for congruent elliptic curves $ny^2=x^3-x$.