Entry 193

Continuing from Entry 192, some more \(G_d\),

$$G_{59} = 2^{-1/4}x,\quad x^3 - 2r x^2 +2 (r^2 - r)x - 2 = 0,\quad r^3-2r^2-1=0\quad$$

and the shared pairs

$$\begin{align}G_{19} &= 2^{-1/4}x,\quad x^3 - 2r x^2 + (r^2 - 5r-2)x - 2 = 0,\quad r=0 \\ G_{379} &= 2^{-1/4}x,\quad x^3 - 2r x^2 + (r^2 - 5r-2)x - 2 = 0,\quad r^3-6r^2-5r-2=0\end{align}$$

and

$$\begin{align}G_{107} &= 2^{-1/4}x,\quad x^3 - 2r x^2 + (r^2 -r-2)x - 2 = 0,\quad r^3-r-4=0\\ G_{139} &= 2^{-1/4}x,\quad x^3 - 2r x^2 + (r^2 - r-2)x - 2 = 0,\quad r^3-r^2-2r-4=0\end{align}$$

The first one has additional context since the real root of \(r^3-2r^2-1=0\) is the supersilver ratio, a cubic analogue of the silver ratio \(1+\sqrt2\).