Congruences for class numbers of \(\mathbb {Q}(\sqrt{\pm 2p})\) when \(p\equiv 3\) \((\text {mod }4)\) is prime

For a prime \(p\equiv 3\) $p\equiv 3$ \((\text {mod }4)\) $(\text{mod}4)$ , let \(h(-8p)\) $h(-8p)$ and h(8p) be the class numbers of \(\mathbb {Q}(\sqrt{-2p})\) $\mathbb{Q}(\sqrt{-2p})$ and \(\mathbb {Q}(\sqrt{2p})\) $\mathbb{Q}(\sqrt{2p})$ , respectively. Let \(\Psi (\xi )\) $\mathrm{\Psi }(\xi )$ be the Hirzebruch sum of a quadratic irrational \(\xi \) $\xi$ . We show that \(h(-8p)\equiv h(8p)\Big (\Psi (2\sqrt{2p})/3-\Psi (\frac{1+\sqrt{2p}}{2})/3\Big )\) $h(-8p)\equiv h(8p)(\mathrm{\Psi }(2\sqrt{2p})/3-\mathrm{\Psi }(\frac{1+\sqrt{2p}}{2})/3)$ \((\text {mod }16)\) $(\text{mod}16)$ . Also, we show that \(h(-8p)\equiv 2\,h(8p)\Psi (2\sqrt{2p})/3\) $h(-8p)\equiv 2h(8p)\mathrm{\Psi }(2\sqrt{2p})/3$ \((\text {mod }8)\) $(\text{mod}8)$ if \(p\equiv 3\) $p\equiv 3$ \((\text {mod }8)\) $(\text{mod}8)$ , and \(h(-8p)\equiv \big (2\,h(8p)\Psi (2\sqrt{2p})/3\big )+4\) $h(-8p)\equiv (2h(8p)\mathrm{\Psi }(2\sqrt{2p})/3)+4$ \((\text {mod }8)\) $(\text{mod}8)$ if \(p\equiv 7\) $p\equiv 7$ \((\text {mod }8)\) $(\text{mod}8)$ .