On the Diophantine equations $d_ 1x^ 2+2^ {2m}d_ 2=y^ n$ and $d_ 1x^ 2+d_ 2=4y^ n$

Abstract:

Let ${d_1},\;{d_2}$ be coprime positive integers, which are squarefree, and let $h$ denote the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt { - {d_1}{d_2}} )$. Let $m,\;n$ be integers such that $m \geqslant 0,\;n > 1$, and $\gcd (n,2h) = 1$. In this paper we prove that if $n \geqslant 8.5 \cdot {10^6}$, then the equations ${d_1}{x^2} + {2^{2m}}{d_2} = {y^n}(2\nmid y)$ and ${d_1}{x^2} + {d_2} = 4{y^n}$ have no positive integer solutions $(x,y)$ with $\gcd (x,y) = 1$.