Entry 20

One of Ramanujan’s talents was spotting elegant special cases of general phenomena. Consider for example $$x=\small+\sqrt{a+\sqrt{a+\sqrt{a-\sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a-\cdots}}}}}}}\tag1$$ where the repeating pattern of the signs is $(+,+,+,-)$. Generally, $x$ will be the root of a $12$th deg factor of a $16$th deg equation. But he found that $$x_1=\small+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\cdots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}=2.7472\dots$$ Note that $x$ is just a root of a quartic. Its other roots are given by the patterns $(+,+,-,+),\; (+,-,+,+),\; (-,+,+,+)$, respectively $$x_2=\small+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\cdots}}}}}}} = \frac{2-\sqrt 5 +\sqrt{15+6\sqrt 5}}{2}=2.5473\dots$$ $$x_3=\small+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\cdots}}}}}}} = \frac{2+\sqrt 5 -\sqrt{15-6\sqrt 5}}{2}=1.4888\dots$$ $$x_4=\small\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}}}}} = \frac{2-\sqrt 5 -\sqrt{15+6\sqrt 5}}{2}=-2.7833\dots$$ This immediately implies that the four roots obey the system with $a=5$, $$\begin{aligned} x_1^2 &= x_2+a\\ x_2^2 &= x_3+a\\ x_3^2 &= x_4+a\\ x_4^2 &= x_1+a\\ \end{aligned}\tag2$$ also studied by Ramanujan. In general, an infinitely nested radical with period length of 4 like $(1)$ $$x = \pm\sqrt{a\pm \sqrt{a\pm \sqrt{a\pm \sqrt{a\pm\dots}}}}$$ and a system of 4 equations like $(2)$ can be expressed as $$x = (((x^2 - a)^2 - a)^2 - a)^2 - a\tag3$$ Expanded out and factored, Ramanujan stated that $(3)$ was a product of 4 quartics, three of which had coefficients in the cubic, $$y^3+3y = 4(1+ay)$$ However, this has a rational factor for the special cases when $a=2,5$ so explains why the radical he found has no cubic irrationalities.