Entry 194

Previous entries discussed \(G_d\) with \(d\equiv 3\,\text{mod}\,8\). For \(d\equiv 7\,\text{mod}\,8\) with odd class number \(h(-d)=\color{blue}n\), then the formula is slightly different

$$G_d = 2^{1/4}x$$

where \(x\) is a root of an algebraic equation of degree \(\color{blue}n\) that is solvable in radicals and a unit constant term.

I. n = 1 $$G_7 = 2^{1/4}x,\quad x-1 = 0\quad $$II. n = 3

$$\begin{align}G_{23} &= 2^{1/4}x,\quad x^3-x-1=0\\ G_{31} &= 2^{1/4}x,\quad x^3-x^2-1=0 \end{align}$$III. n = 5

$$\begin{align}G_{47} &= 2^{1/4}x,\quad x^5 - x^3 - 2x^2 - 2x - 1 = 0\\ G_{79} &= 2^{1/4}x,\quad x^5 - 3x^4 + 2x^3 - x^2 + x -1=0\\G_{103} &= 2^{1/4}x,\quad x^5 - x^4 - 3x^3 - 3x^2 - 2x - 1 = 0\\ G_{127} &= 2^{1/4}x,\quad x^5 - 3x^4 - x^3 + 2x^2 + x - 1 = 0 \end{align}$$

\(G_{23}\) and \(G_{31}\) were known to Ramanujan and their \(x\) involve the plastic ratio and supergolden ratio, respectively. Ramanujan also solved two quintics in radicals with \(d = 47, 79\) so he probably found \(G_{47}\) and \(G_{79}\) though I'm not sure for \(G_{103}\) and \(G_{127}\).