I was browsing in Plouffe's inverter and found that $\frac{\exp(\pi)\ln(3)}{\ln(2)}$ is very nearly $\frac{159}{5}.$ The continued fraction is
[31, 1, 4, 12029125, ...].
Is this the same magic as $\exp(\pi \sqrt{163})$?
I was browsing in Plouffe's inverter and found that $\frac{\exp(\pi)\ln(3)}{\ln(2)}$ is very nearly $\frac{159}{5}.$ The continued fraction is [31, 1, 4, 12029125, ...]. Is this the same magic as $\exp(\pi \sqrt{163})$? 


The trick with the modular invariant $j$ is for $\pi\sqrt D$ only (as $j(\pi\sqrt{D})$ is rational). Your value is not the exponential of a CMpoint in the upper halfplane, so nothing to do with modularity. This kind of experimental discoveries already exists in the literature; see, for example, [C. Schneider, R. Pemantle. When is 0.999... equal to 1?. Amer. Math. Monthly 114 (2007) 344350.]. 

