Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees

TL;DR

This paper introduces a method using 2-adic valuation trees to analyze and solve specific quadratic and cubic Diophantine equations, identifying conditions for infinite solutions based on the structure of these trees.

Contribution

The paper presents a novel approach employing 2-adic valuation trees to analyze solutions of quadratic and cubic Diophantine equations, including criteria for infinite solutions.

Findings

Identification of conditions for infinite valuation trees

Application of valuation trees to specific Diophantine equations

Characterization of solutions based on 2-adic valuations

Abstract

For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in \mathbb{Z}^+$, the numbers $x^3+D$ will generate infinite valuation trees, which lead to infinite solutions to the above Diophantine equations.