On the Diophantine equation Cx²+p₁2m₁p₂2m₂⋯ pₖ2mₖ=yq

Let $C>3$ be an odd square-free integer, q an odd prime not dividing the class number of Q(√-C), and p₁,,pₖ distinct odd primes not dividing C. We describe all solutions of Cx²+p₁2m₁⋯ pₖ2mₖ=yq in positive integers with (x,y)=1 and y odd. Writing N=p₁m₁⋯ pₖmₖ, every solution comes from a unique factorization x√ C+N√-1=±(u√ C± v√-1)q, or from a half-integral analogue when $q=3$, in which each pᵢ≠ q divides exactly one of v and $N/v$. Using primitive divisors of Lehmer sequences, we show that for q≥5 some pᵢ satisfies pᵢ≡-Cpᵢ q; in particular, there is no solution when q≥max(5,p₁+2,,pₖ+2). For $q=3$ the equation reduces to four explicit auxiliary equations.