TITLE:
Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited
AUTHORS:
Werner Hürlimann
KEYWORDS:
Sum of Squares, Ternary Quadratic Form, Theta Function, Hurwitz Three-Squares Formula, Congruent Number, Weak Birch-Swinnerton-Dyer Conjecture
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.5,
April
16,
2015
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.