On a class of Lebesgue-Ramanujan-Nagell equations

Abstract

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for prime $n$, under the condition $n\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field $\mathbb{Q}(\sqrt{-cd})$. We then completely solve this equation for both $c$ and $d$ primes under the assumption that $\gcd(n, h(-cd))=1$. We also completely solve this equation for $c=1$ and $d\equiv1 \pmod 4$, under the condition $\gcd(n, h(-d))=1$. For some fixed values of $c$ and $d$, we derive some results concerning the solvability of this equation.