Unsolvability of the equation \(x^n-y^2=az^m\) with small greatest common divisor \(\text{gcd}(n,h(-a))\) (Q1383686)

The author deals with equations \[ x^n-y^2= az^{2m} \tag{1} \] in rational integers \(x,y,z\) where \(n,m\) and \(a\) are positive integers and where \(n\geq 2\). One of his results is as follows. Denote by \(h(-a)\) the class number of \({\mathbb Q}(\sqrt{-a})\). Then there are only finitely many integers \(a\) with \(a>0\), \(a\not\equiv 3\pmod 4\) and \(n/\text{gcd} (n,h(-a))\geq\max (6, 14-2m)\) such that equation (1) is solvable in integers \(x,y,z\) with \(\text{gcd} (x,y,z)=1\). His proof uses Faltings' theorem that algebraic curves of genus \(\geq 2\) have only finitely many rational points. Under the assumption of Schinzel's conjecture that an irreducible polynomial \(P(X)\in{\mathbb Z}[X]\) assumes infinitely many prime numbers, the author deduces from his result mentioned above that an arithmetic progression \(\alpha x+\beta\) satisfying some very specific technical conditions depending on \(n\) contains infinitely many primes \(p\) with \(h(-p)\equiv 0\pmod n\).