Ternary quadratic forms representing the same integers

Abstract

In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this paper, we prove the existence of pairs of ternary quadratic forms representing the same integers which are not contained in Kaplansky’s list.