On the Gauss E ϒPHKA theorem and some allied inequalities

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: 9x2+16y2+36z2+16yz+4xz+8xy9x2+16y2+36z2+16yz+4xz+8xy and 9x2+17y2+32z2−8yz+8xz+6xy9x2+17y2+32z2−8yz+8xz+6xy. We also discuss three nontrivial analogues of the Gauss E ϒPHKA theorem. The technique used can be applied to all known spinor regular ternary quadratic forms.