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

Abstract

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

The 2-adic valuation of the general degree-2 polynomial in 2 variables

“…From the point of view of application, the 2-adic valuation tree is used to solve quadratic and cubic diophantine equations in [5]. Apart from this, some other special cases are studied in [1,7,12].…”