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

For fixed integers $D\ge 0$ and $c\ge 3$, 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 $D\in {\mathbb{Z}}^{+}$, the numbers $x^3+D$ will generate infinite valuation trees, which lead to infinite solutions to the above Diophantine equations.