Primitive Representations of Integers by $x^3+y^3+2z^3$

Abstract

A well-known open problem is to show that the cubic form $x^3+y^3+2z^3$ represents all integers. An obvious variant of this problem is whether every integer can be {\em primitively} represented by $x^3+y^3+2z^3$. In other words, given an integer $n$, are there coprime integers $x$, $y$, $z$ such that $x^3+y^3+2z^3=n$? In this note we answer this variant question negatively. Indeed, we use cubic reciprocity to show that for every integral solution to $x^3+y^3+2z^3=8^m$,the unknowns $x$, $y$, $z$ are divisible by $2^m$.

This paper has been withdrawn by the author. Roger Heath-Brown has pointed out to me that the paper duplicates his result found in: "The density of zeros of forms for which weak approximation fails", Math. Comp. 59 (1992), 613-623