## The Brioschi quintic and the Rogers-Ramanujan continued fraction

Given Ramanujan’s constant, $e^{\pi\sqrt{163}} \approx Q+743.9999999999992\dots$

where $Q = 640320^3$, why do we know, in advance, that the quintic, $w^5-10(\frac{1}{1728+Q})w^3+45(\frac{1}{1728+Q})^2w-(\frac{1}{1728+Q})^2 = 0$

is solvable in radicals?  The answer is this: The general quintic can be transformed in radicals to the one-parameter form, $w^5-10cw^3+45c^2w-c^2 = 0$

called the Brioschi quintic.  Whether reducible or not, if it is solvable in radicals and c is rational, then it can be shown c must have the form, $c = 1/(1728-t)$

where, $t = \frac{(u^2+10u+5)^3}{u}$

for some radical u. (For example, u = 1 will yield an irreducible though solvable quintic.)  But it seems Nature likes to recycle polynomials as this is also one of the many formulas for the j-function $j(\tau)$, namely, $j(\tau) = \frac{(v^2+10v+5)^3}{v}$

where, $v = \Big(\frac{\sqrt{5}\,\eta(5\tau)}{\eta(\tau)}\Big)^6$

and $\eta(\tau)$ is the Dedekind eta function.  However, if we let, $v = \frac{-125r^5}{r^{10}+11r^5-1}$

then we get the more well-known j-function formula, $j(\tau) = \frac{-(r^{20}-288r^{15}+494r^{10}+228r^5+1)^3}{r^5(r^{10}+11r^5-1)^5}$

where the numerator and denominator are polynomial invariants of the icosahedron, a Platonic solid wherein one can find pentagons (which, of course, has 5 sides).  Perhaps not surprisingly, r is given by the eta quotient, $r^{-1}-r = \frac{\eta(\tau/5)}{\eta(5\tau)}+1$

But one can also express r using the beautiful Rogers-Ramanujan continued fraction. Let, $q = e^{2 \pi i \tau}$

then, $r = r(\tau) = \cfrac{q^{1/5}}{1+ \cfrac{q}{1 + \cfrac{q^2}{1+ \cfrac{q^3}{1 + \ddots}}}}$

One of the simplest cases is, $r(\sqrt{-1}) = 5^{1/4}\sqrt{\frac{1+\sqrt{5}}{2}}-\frac{1+\sqrt{5}}{2}= \cfrac{e^{-2\pi/5}}{1+ \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+ \cfrac{e^{-6\pi}}{1 + \ddots}}}} \approx 0.284079$

which was communicated by Ramanujan to Hardy in his famous letter.

Interesting, isn’t it?

For a related topic, kindly read “Ramanujan’s Continued Fractions and the Platonic Solids“.