Entry 137

This continues Entry 136. Recall the function $$\beta(n) = \Big(\sqrt{n^2-2}+\sqrt{n^2-1}\Big)^2$$ and the examples of $n$ which were quartic roots. It turns out these $n$ have additional properties which yield fundamental units $U_k$ though I don't know why.

For $p=31$, let $n_{\color{red}\pm} = 2(1+\sqrt2)^2\Big(1+3\sqrt2\color{red}{\pm}2\sqrt{1+4\sqrt2}\Big)$ or the two real roots of the quartic. Then $$\frac{\beta(n_{+})}{\beta(n_{-})} = U_{31}\sqrt{U_{62}} = (1520+273\sqrt{31})\,(4\sqrt2+\sqrt{31})$$ For $p=47$, let $n_{\color{red}\pm} = 2(1+\sqrt2)^3\Big(9\color{red}{\pm}2\sqrt{9+8\sqrt2}\Big)$ or again the two real roots. Then $$\frac{\beta(n_{+})}{\beta(n_{-})} = U_{47}\sqrt{U_{94}} = (48+7\sqrt{47})\,(732\sqrt2+151\sqrt{47})$$ and so on for $p = 31, 47, 79, 191, 239, 431$.