Entry 172

There are many fundamental \(d=4m\) with class number \(8\), though only seven are prime namely \(p = 41, 113, 137, 313, 337, 457, 577\). Their Ramanujan \(G\)-functions are

$$\begin{align}G_{41} &= \left(\frac{x+\sqrt{x^2-4}}2\right)^{1/2} =\sqrt{\frac{x-2}4}+\sqrt{\frac{x+2}4} \\ G_{113} &=\left(\frac{y+\sqrt{y^2-4}}2\right)^{1/2} =\sqrt{\frac{y-2}4}+\sqrt{\frac{y+2}4} \end{align}$$ where $$x = \left(\frac{5+\sqrt{41}}4\right) \left(1+\sqrt{\frac{-5+\sqrt{41}}8}\right)\\ y = \left(\frac{9+\sqrt{113}}4\right) \left(1+\sqrt{\frac{-7+\sqrt{113}}2}\right)$$ and similarly for the other \(p\), though the quartic roots get more complicated.