[gjm11]

Very, very, very hard.

You almost certainly know that already, so here are a couple more remarks just for fun:

One useful technique when making pseudorandom number generators is to take two not-so-good RNGs and combine their outputs (e.g., adding them or XORing them). If the two RNGs have different enough "structure", the combined generator can have much better statistical properties than either of its two components. The fact that it's much harder to prove anything about pi+e than about pi or e individually is rather like that.

But even proving that pi is irrational is highly nontrivial. (Proving that e is irrational, on the other hand, is a fairly easy exercise. Sketch: suppose e = p/q; then q!e is an integer; but q!e is the sum of a series whose terms are initially integers and then abruptly positive numbers small enough that their sum has to be between 0 and 1; contradiction.)

[jtolle]

How awesome would it be if pi + e were found to be rational? Maybe everything starts cancelling out past the zillionth digit...