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arXiv · 2105.03352

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

Maila Brucal-Hallare·Eva G. Goedhart·Ryan Max Riley·Vaishavi Sharma·Bianca Thompson
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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.

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Maila Brucal-Hallare, Eva G. Goedhart, Ryan Max Riley, Vaishavi Sharma, Bianca Thompson. 2021-05-07. Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees. https://arxiv.org/abs/2105.03352

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