Wrong part of 'Reference' deleted. (K. Mahler, 1953) AMM, Problem 10630, Richard Strong should changed to (K. Mahler, 1953) Remarks from PEN book: This is a deep theorem in transcendental number theory. Note that it follows from this result that $ \pi$ is irrational! In fact, it's known that for suciently large q, the exponent 42 can be replaced by 30. Here is a similar result due to A. Baker: for any rationals $ \frac{p}{q}$, one has \[ \left\vert\ln 2-\frac{p}{q}\right\vert\ge 10^{-100000}q^{-12.5}.\]

Why do we need such "deep theorems" into PEN? Probably none of our readers can prove it, and it is not elementary either?

Peter wrote: Why do we need such "deep theorems" into PEN? Probably none of our readers can prove it, and it is not elementary either? When I saw it first, it was very impressive for me. So that's why I include it into PEN. As Peter mentioned, the proof is highly non-elementary, so let's remove those difficult problems. (Peter, but let's keep the latex source of those deleted problems for the cases.)