Conway's light on the shadow of Mordell

TL;DR

This paper explores the shadow solutions of the Mordell Diophantine equation using Conway's topograph and Pell's equation, revealing growth patterns via Lyapunov functions and extending concepts from Markov triples.

Contribution

It introduces a novel approach to analyze Mordell triples through shadow solutions, connecting them with Conway's topograph and Pell's equation solutions.

Findings

Shadow solutions of Mordell triples are explicitly described.

Growth of solutions along paths is characterized by Lyapunov functions.

Connections between Mordell solutions, Conway topograph, and Pell's equation are established.

Abstract

Recently Valentin Ovsienko introduced a ``shadow" version of the celebrated Markov triples as the solutions of certain version of Markov equation over dual numbers. We will discuss similar question for the Mordell Diophantine equation $$ X^2+Y^2+Z^2=2XYZ+1. $$ We will see that the shadows of the special solution $(1,1,1)$ can be described using the original Conway topograph of the values of binary quadratic forms. The shadows of other Mordell triples will be written explicitly in terms of the corresponding solutions of Pell's equation. Their growth along the paths on the Conway topograph is described in terms of the Lyapunov function of the Euclid tree.