On the class number of \({\mathbb {Q}}(\sqrt{(2^ra)^{2}- y^p})\)

Let p be an odd prime, and let \(r>1\) $r>1$ and a be positive integers such that \(2 \not \mid a\) $2\nmid a$ . In this article, we prove the existence of infinitely many integers y such that if \((y,a) = 1\) $(y,a)=1$ , then p divides class number of the imaginary quadratic field \({\mathbb {Q}}(\sqrt{(2^ra)^{2} - y^p})\) $\mathbb{Q}(\sqrt{{(2^{r}a)}^2-y^p})$ .