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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 $\exp(\pi \sqrt{163})$?

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There doesn't seem to be a straightforward relationship with class field theory. See also: xkcd.com/217 –  S. Carnahan May 5 '10 at 1:03
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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 CM-point 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) 344-350.].

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