# Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the complete elliptic integral of the first kind $K(k_\color{blue}m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\tfrac12}$.

Conjecture: "The equalities hold for real $N>1$. But for any integer $N>1$, then the hypergeometric function $_2F_1(z)$ and its argument $z$ are algebraic numbers."

I. For $a=\tfrac14$ and $\tau=N\sqrt{-4}$

\begin{aligned}\,_2F_1\left(\tfrac14,\tfrac14;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{(1+\sqrt2)}{4\sqrt2}\frac{_2F_1\left(\tfrac12,\tfrac12;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}4)}\\[2mm] \end{aligned}\tag1 $$w=\frac{16}{16+\Big(\tfrac{\eta(\tau/4)}{\eta(\tau)}\Big)^8}$$

Example: If $\tau=2\sqrt{-4}$, then, $$_2F_1\left(\tfrac14,\tfrac14;\tfrac12;\,9(11-8\sqrt2)^2\right)=\tfrac{3}{4\sqrt2}(1+\sqrt2)$$

II. For $a=\tfrac16$ and $\tau=N\sqrt{-3}$

\begin{aligned}\,_2F_1\left(\tfrac16,\tfrac13;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{1}{27^{1/4}}\frac{_2F_1\left(\tfrac13,\tfrac23;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}3)}\\[2mm] \end{aligned}\tag2 $$w=\frac{27}{27+\Big(\tfrac{\eta(\tau/3)}{\eta(\tau)}\Big)^{12}}$$ Example: If $\tau=2\sqrt{-3}$, then, $$_2F_1\left(\tfrac16,\tfrac13;\tfrac12;\,\tfrac{25}{27}\right)=\tfrac{3\sqrt3}{4}$$

III. For $a=\tfrac18$ and $\tau=N\sqrt{-2}$

\begin{aligned}\,_2F_1\left(\tfrac18,\tfrac38;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{\sqrt{1+\sqrt2}}{2^{7/4}}\frac{_2F_1\left(\tfrac14,\tfrac34;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}2)}\\[2mm] \end{aligned}\tag3 $$w=\frac{64}{64+\Big(\tfrac{\eta(\tau/2)}{\eta(\tau)}\Big)^{24}}$$Example: If $\tau=3\sqrt{-2}$, then, $$_2F_1\left(\tfrac18,\tfrac38;\tfrac12;\tfrac{2400}{2401}\right)=\tfrac{2\sqrt7}{3}$$

IV. For $a=\tfrac1{12}$ and $\tau=N\sqrt{-1}$

\begin{aligned}\,_2F_1\left(\tfrac1{12},\tfrac5{12};\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{1}{12^{1/4}}\frac{_2F_1\left(\tfrac16,\tfrac56;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}1)}\\[2mm] \end{aligned}\tag4 $$\frac{432}{w(1-w)} =j(\tau)$$ Example: If $\tau=2\sqrt{-1}$, then, $$_2F_1\left(\tfrac1{12},\tfrac5{12};\tfrac12;\tfrac{1323}{1331}\right)=\tfrac{3\,\sqrt{11}}{4}$$

This is a highly compactified version of the results by Zucker and Joyce in "Special values of the hypergeometric series II, III" but I used a common form for the argument $z = (1-2w)^2$ as well as similar eta quotients to better illustrate their affinity. However, I only derived this empirically.

Q: How do we rigorously prove the four conjectures?