On the shortest open cubic equations

TL;DR

This paper employs cubic reciprocity to prove the non-existence of integer solutions for a specific cubic equation, identifying it as the shortest unresolved cubic equation prior to this work.

Contribution

It introduces a novel proof technique using cubic reciprocity to resolve the solvability of a previously open shortest cubic equation.

Findings

Proves that 7x^3+2y^3=3z^2+1 has no integer solutions

Identifies the equation as the shortest unresolved cubic equation before this study

Provides a list of new shortest open cubic equations

Abstract

We use cubic reciprocity to prove that the equation $7x^3+2y^3=3z^2+1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations.