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- Using the Gelfond-Schneider theorem show that if is transcendental then is transcendental
- If are positive integers such that and is even, show that is not a square
- This theorem was being discussed at a seminar many years ago as was the transcendence of . When we were told that it was an obvious consequence that is transcendental we thought that the speaker was joking. When he showed how it followed, there was a moment’s silence and then applause – one of those moments when one realises that mathematics is such a wonderful subject.
You can find an elementary proof (not using Gelfond-Schneider) of the transcendence of and here
It’s interesting to note that and are very close. In fact . See here
- I came across this problem when looking through some old papers. I have no idea where it comes from (an old Mathematical Olympiad problem maybe??) or what the solution was.