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

TL;DR

This paper investigates the primitive representation of integers by the form x^3 + y^3 + 2z^3 and demonstrates that not all integers can be primitively represented, using cubic reciprocity to establish divisibility constraints.

Contribution

It provides a negative answer to the question of primitive representability for the form x^3 + y^3 + 2z^3, showing divisibility conditions for solutions.

Findings

Not all integers are primitively represented by x^3 + y^3 + 2z^3.

Solutions to x^3 + y^3 + 2z^3 = 8^m are divisible by 2^m.

Cubic reciprocity is used to derive divisibility constraints.

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 $2^m$.