All congruent numbers less than 10 000
The main result of this paper is the following: If \(n\) is a square-free positive integer less than 10000, then \(n\) is a congruent number if and only if \(L(E_ n,1)=0\). Here \(E_ n\) is the elliptic curve represented by \(y^ 2= x^ 3- n^ 2 x\), and \(L(E_ n,s)\) is the Hasse-Weil \(L\)- function of \(E_ n\). Using results of \textit{J. B. Tunnell} [Invent. Math. 72, 323-334 (1983; Zbl 0515.10013)] and \textit{B. H. Gross} and \textit{D. Zagier} [C. R. Acad. Sci., Paris, Sér. I 297, 85-87 (1983; Zbl 0538.14023)] the authors are able to list all congruent numbers below 10000.
scientific article; zbMATH DE number 556447
scientific article