On the Diophantine equation $Cx^2+p^{2m}=y^q$

Abstract

We classify the primitive positive solutions of $Cx^2+p^{2m}=y^q$ under the hypotheses that $C>3$ is odd and square-free, $p,q$ are odd primes, $p\nmid C$, $q\nmid h(-C)$, and $y$ is odd. The solutions form six explicit families: one for arbitrary $q$ and five exceptional families with $q=3$, three of which arise from half-integral elements when $C\equiv3\pmod8$. The proof combines ideal factorization with Lehmer sequences and shows that representations with second coordinate $v=p^r$, $r>0$, contribute only when $q=3$.