A generic example is ${}\_2 F\_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric function ${}\_m F\_n$ takes rational values?


The references to the literature where a lot of such examples listed are apreciated very much too.



at which rational points does the Hypergeometric function take rational values
There are theorems that give conditions for the set to be finite even when "rational" is replaced by "algebraic". See the work of Paula Cohen Tretkoff. E.g the following paper is a survey and Theorem 2 is about this:


<http://www.math.tamu.edu/~ptretkoff/martinpub_final.pdf>


