Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Abstract

Bell's theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms 2 2 2 x by cz + + with { } , 1, 2, 4, 8 b c ∈. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell's theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell's theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel's congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number ( ) 3 mod 8 n ≡ is not congruent.