On The Gauss EYPHKA Theorem And Some Allied Inequalities

TL;DR

This paper explores the properties of specific ternary quadratic forms using classical formulas, introduces analogues of the Gauss EYPHKA theorem, and demonstrates a technique applicable to all known spinor regular forms.

Contribution

It presents new analogues of the Gauss EYPHKA theorem and applies classical formulas to analyze representation properties of certain quadratic forms.

Findings

Derived representation properties of two JP forms of Kaplansky.

Introduced three nontrivial analogues of the Gauss EYPHKA theorem.

Technique applicable to all known spinor regular ternary quadratic forms.

Abstract

We use the 1907 Hurwitz formula along with the Jacobi triple product identity to understand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy and 9x^2+ 17y^2 +32z^2 -8yz+ 8xz + 6xy. We also discuss three nontrivial analogues of the Gauss EYPHKA theorem. The technique used can be applied to all known spinor regular ternary quadratic forms.