Entry 90

The modular lambda function given by Mathematica as ModularLambda[tau] can solve $$\frac{_2F_1\big(\tfrac12,\tfrac12,1,1-\lambda(\tau)\big)}{_2F_1\big(\tfrac12,\tfrac12,1,\lambda(\tau)\big)}  = -\tau\sqrt{-1}$$ In the previous entry, we used the tribonacci constant for the case \(\tau=\sqrt{-11}\). More generally, to solve the higher Heegner numbers \(d = 11,19,43,67,163\) we can employ a slightly different method. For example, $$\frac{_2F_1\big(\tfrac12,\tfrac12,1,1-x\big)}{_2F_1\big(\tfrac12,\tfrac12,1,x\big)}  = \sqrt{163}$$ This also has an exact solution expressible in radicals \(x \approx 6.094\times10^{-17}\) as the smaller root of the quadratic $$16x^2-16x+1=y$$ and \(y\) is the real root of the cubic $$y^3+\frac{640320^3}{256}y-\frac{640320^3}{256}=0$$ The big number should be familiar and appears in Ramanujan's constant $$e^{\pi\sqrt{163}} = 640320^3 + 743.99999999999925\dots$$ For the other Heegner numbers \(d\), one just replaces the big number with the cube nearest to \(e^{\pi\sqrt{d}}-744\), like \(5280^3\) for \(\sqrt{67}\) and so on.