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

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 indecompos-able positive definite binary quadratic form exists. We show that the two problems are equivalent. Recently Borwein and Choi [Borwein and Choi 00], and independently Le [Le 98], have shown that the Diophan-tine equation xy + yz + zx = n has solutions x, y, z with x, y, z ≥ 1 for all natural numbers n with the excep