On a class of Lebesgue-Ramanujan-Nagell equations

Abstract

We deeply investigate the Diophantine equation \(cx^2+d^{2m+1}=2y^n\) in integers \(x, y\ge 1, m\ge 0\) and \(n\ge 3\), where c and d are coprime positive integers satisfying \(cd\not \equiv 3 \pmod 4\). We first solve this equation for prime n under the condition \(\gcd (n, h(-cd))=1\), where \(h(-cd)\) denotes the class number of the imaginary quadratic field \({\mathbb {Q}}(\sqrt{-cd})\). We then completely solve this equation for both c and d primes under the assumption \(\gcd (n, h(-cd))=1\). We also completely solve this equation for \(c=1\) and \(d\equiv 1 \pmod 4\) 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.