A note on the Lebesgue–Ljunggren–Nagell equation $ax^2+b^{2m}=4y^n$

Abstract

Let a, b be fixed positive integers such that $2\not \mid {ab}$, $\gcd (a,b)=1$ and a is square-free, and let $h(-a)$ denote the class number of the imaginary quadratic field ${\mathbb {Q}} (\sqrt{-a})$. Further, let n be a fixed prime satisfying $n>3$, $n\not \mid h(-a)$ and $n\not \mid b$. In this paper, we investigate the solvability of the equation $ax^2+b^{2m}=4y^n$, $x,y,m\in {\mathbb {N}}$, $\gcd (x,y)=1$. We prove that if every prime divisor p of b satisfies $p\not \equiv \pm 1 \pmod {n}$, then the equation has no solutions (x, y, m). Moreover, under the assumption that b is an odd prime, there exist only finitely many triples (a, b, n) for which the equation has a solution (x, y, m) with $m>1$.