It's not really "of course", and I don't think we have such a theorem in general. But in this case, I believe the fact that it's not an integer follows from th

It's not really "of course", and I don't think we have such a theorem in general. But in this case, I believe the fact that it's not an integer follows from the same theorem that says it's very close to an integer. See eg https://math.stackexchange.com/questions/4544/why-is-e-pi-sqrt163-almost-an-integer https://math.stackexchange.com/questions/4544/why-is-e-pi-sq... Basically e^(sqrt(163)*pi) is the leading term in a Laurent series for an integer, and the other (non-integer) terms are really small but not zero.