# Mathematics Weblog

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## 2 unrelated problems

Sunday 12 December 2004 at 2:26 pm | In Articles | 2 Comments- 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

__Notes__

- 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.

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Here’s a solution.

Suppose that x^2 = a^2 – a – b.

Then one finds that

4x^2 + 1 = (2a+2b-1)(2a-2b-1)

We are assuming that a & b have the

same parity, and that a>b are positive.

Thus 2a+2b-1 and 2a-2b-1 are both positive

integers of the form -1 mod 4. It’s easy to

see that such numbers have a prime factor

p of the form -1 mod 4. Thus 4x^2+1 has

a prime factor of the form p = -1 mod 4.

Now we are done, since all prime factors of

4x^2+1 are of the form 1 mod 4.

(This follows for example by quadratic

reciprocity).

Comment by Frank — Friday 18 February 2005 7:44 am #

Wonderful, thank you! Now why didn’t I see that 4x

^{2}+ 1 = (2a+2b-1)(2a-2b-1)?Comment by Steve — Friday 18 February 2005 2:21 pm #