Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the shortest open cubic equations

Bogdan Grechuk, Ashleigh Ratcliffe

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.

On the shortest open cubic equations

Abstract

We use cubic reciprocity to prove that the equation 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.

Paper Structure

This paper contains 4 sections, 5 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Eisenstein integers and cubic reciprocity
  3. The resolution of the shortest open equation
  4. The new shortest open equations

Key Result

Proposition 2.1

modern

Theorems & Definitions (7)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof