The Diophantine Equation $ xy + yz + xz = n $ and Indecomposable Binary Quadratic Forms

Abstract

There are 18 (and possibly 19) integers that are not of the form $ xy + yz + xz $ with positive integers $x, y, z$. The same 18 integers appear as exceptional discriminants for which no indecomposable positive definite binary quadratic form exists. We show that the two problems are equivalent.