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

Werner Hürlimann^*

Swiss Mathematical Society, Fribourg, Switzerland.

DOI: 10.4236/apm.2015.55027                

Abstract

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x²+by²+cz² with b,c {1,2,4,8}. 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 n 3(mod 8) is not congruent.

Keywords

Sum of Squares, Ternary Quadratic Form, Theta Function, Hurwitz Three-Squares Formula, Congruent Number, Weak Birch-Swinnerton-Dyer Conjecture

Conflicts of Interest

The authors declare no conflicts of interest.

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