Entry 173

There are many in the set \(d=4m\) with class number \(8\). When \(m\) is a prime or a semiprime (a product of two primes) like \(m =3p\), then it may be well-behaved. For this set, there are only three, namely \(p = 23, 47, 71\), thus \(m = 3p = 69, 141, 213\). Ramanujan found the radicals below and the \(G\)-function have a common form

$$G_{3p} = U_{p}^{1/24}\,U_{3p}^{1/16}\,x_{p}^{1/2}$$

with fundamental unit \(U_n\) and where \(x_{p}^2\) is a root of a unit quartic

$$\begin{align}G_{69} &= \left(\frac{5+\sqrt{23}}{\sqrt2}\right)^{1/12} \left(\frac{3\sqrt3+\sqrt{23}}2\right)^{1/8} \left(\sqrt{\frac{2+3\sqrt3}4}+\sqrt{\frac{6+3\sqrt3}4}\right)^{1/2}\\ G_{141} &= \left(\frac{7+\sqrt{47}}{\sqrt2}\right)^{1/12}\; \big(4\sqrt3+\sqrt{47}\big)^{1/8}\; \left(\sqrt{\frac{14+9\sqrt3}4}+\sqrt{\frac{18+9\sqrt3}4}\right)^{1/2}\\ G_{213} &= \left(\frac{59+7\sqrt{71}}{\sqrt2}\right)^{1/12} \left(\frac{5\sqrt3+\sqrt{71}}2\right)^{1/8} \left(\sqrt{\frac{19+12\sqrt3}2}+\sqrt{\frac{21+12\sqrt3}2}\right)^{1/2} \end{align}$$ How Ramanujan found these is unknown as it is uncertain if he was aware of class field theory. Note also that without the factor \(3\), then \(d=23,47,71\) are the smallest \(d\) with class number \(h(-d) = 3,5,7\), respectively, while \(h(-3d) = 8\).

P.S. Checking \(h(-3d) = 16\), one finds the only primes are \(d=167,191,239,383,311\) which are the smallest \(d\) with class number \(h(-d) = 11,13,15,17,19\), respectively. Makes me wonder if their \(G_{3d}\) would be analogous.