Bill Gosper <billgosper@gmail.com> writes:
[...] Are there more spectacular e^(\pi algebraic)?
Do you mean "more spectacular than \sqrt{163}" or "other spectacular examples"? For the former, none that I know. For the latter, yes, all with the algebraic number being sqrt(d) for some rational d. The further integral examples d=67 and (barely) 43 are well-known; since the cube root of j is also a modular function (and some other things work out nicely), you can also use 163/9, 67/3, and (barely) 43/3. [I've suggested that a CM mile should be exactly exp(Pi*sqrt(67)/3) feet long. :-)] There's also the known example of d=58, and here it also 58/4 and 58/16 that yield near-integers. There are also some even less well-known examples like exp(Pi*sqrt(89/3)) and its cube root 300.000155555... Some of these also have small powers that are nearly integral but not as close, e.g. exp(Pi*sqrt(163)*4/3) = 168107956062137200957439.99999965975... NDE