Entry 175

For discriminant \(d=4m\) with class number \(8\) and semiprime \(m\): 

If \(m =5p\), then distinguish between \(p \equiv 1\,\text{mod}\, 8\) and \(p \equiv 5\,\text{mod}\, 8\). 

If \(m =7p\), then distinguish between \(p \equiv 3\,\text{mod}\, 8\) and \(p \equiv 7\,\text{mod}\, 8\). 

The semiprime \(m =5p\) was in the previous entry. For \(m =7p\), there are only four and Ramanujan found the radicals below. 

For the 1st case, \(p = 11, 43\), thus \(m = 7p = 77,\, 301\)  

$$\begin{align}G_{77} &= \big(8+3\sqrt7\big)^{1/8}\left(\frac{\sqrt7+\sqrt{11}}2\right)^{1/8} \left(\sqrt{\frac{2+\sqrt{11}}4}+\sqrt{\frac{6+\sqrt{11}}4}\right)^{1/2}\\ G_{301} &= \big(8+3\sqrt7\big)^{1/8}\left(\frac{57\sqrt7+23\sqrt{43}}2\right)^{1/8} \left(\sqrt{\frac{42+7\sqrt{43}}4}+\sqrt{\frac{46+7\sqrt{43}}4}\right)^{1/2}\end{align}$$

For the 2nd case, \(p = 31, 79\), thus \(m = 7p = 217,\, 553\) 

$$\begin{align}G_{217} &= \left(\sqrt{x_1-\tfrac12}+\sqrt{x_1+\tfrac12}\right)^{1/2} \left(\sqrt{y_1-\tfrac12}+\sqrt{y_1+\tfrac12}\right)^{1/2}\\ G_{553} &= \left(\sqrt{x_2-\tfrac12}+\sqrt{x_2+\tfrac12}\right)^{1/2} \left(\sqrt{y_2-\tfrac12}+\sqrt{y_2+\tfrac12}\right)^{1/2} \quad\end{align}$$ where $$x_1=\frac{10+4\sqrt{7}}2,\quad y_1=\frac{14+5\sqrt{7}}4$$ $$x_2=\frac{142+16\sqrt{79}}2,\quad y_2=\frac{98+11\sqrt{79}}4$$

But it seems not noticed that a fundamental unit is imbedded in these radicals as

$$x_1+2y_1 = \frac32(8+3\sqrt{7}) = \frac32\left(\frac{3+\sqrt7}{\sqrt2}\right)^2 =\frac32\,U_7$$

$$x_2+2y_2 = \frac32(80+9\sqrt{79}) = \frac32\left(\frac{9+\sqrt{79}}{\sqrt2}\right)^2 = \frac32\,U_{79}$$