Research Information System
MISKOLC MATHEMATICAL NOTES, vol.27, no.1, pp.283-290, 2026 (SCI-Expanded, Scopus)
Let $d$ be a fixed positive integer with $d>3$ is square free, and let $h(-d)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Further, let $p$ and $q$ be distinct odd primes such that $p>3$ and $p\nmid h(-d)$. In this paper, we give a sufficient and necessary condition for the Lebesgue-Nagell equation $(*)$ $dx^2+p^{2m}q^{2n}=4y^p$ to have positive integer solutions $(x,y,m,n)$ with $\gcd(x,y)=1$. It can be seen from this condition that if $q\not\equiv \pm 1 \pmod{2p}$, then $(*)$ has no positive integer solutions $(x,y,m,n)$ with $\gcd(x,y)=1$.