Equations of the Cayley Surface

TL;DR

This paper explores integer solutions to Cayley's cubic equation, develops recurrence-based solution families, and compares these solutions to Markov numbers, providing new formulas and insights into their properties.

Contribution

It introduces new solution formulas for Cayley's cubic equation and analyzes their similarities and differences with Markov numbers.

Findings

Infinite families of solutions constructed from recurrence relations.

New formulas for solutions analogous to Markov numbers.

Insights into the structural differences between Cayley's solutions and Markov numbers.

Abstract

In this note we study the integer solutions of Cayley's cubic equation. We find infinite families of solutions built from recurrence relations. We use these solutions to solve certain general Pell equations. We also show the similarities and differences to Markov numbers. In particular we introduce new formulae for the solutions to Cayley's cubic equation in analogy with Markov numbers and discuss their distinctions.