I have been meaning to do this for a while …
If is rational then and where and are the smallest nominator and denominator possible, i.e, in the lowest terms.
Hence and so must be even (as two odd numbers multiplied always give an odd number – let and be even numbers then ). Hence .
But we also have and thus and and so is even.
So we have and sharing a common factor of 2, so they cannot be the nominator and denominator of the fraction in the lowest terms.
But if and are both even then they share a common factor of 2 and : implying that and , an obvious contradiction: hence cannot be a rational.
Update: I have made the final step of this shorter and clearer.
Further update: I have been told (see comments below) I would have been better sticking with a clearer version of the original ending ie., we state that are the lowest terms. Then, plainly as they have a factor in common (2) they cannot be the lowest terms and so we have a contradiction. Would be great if someone could explain why we cannot use the contradiction.
And another update: Should have stuck with the original explanation – which I have now restored in a hopefully clearer way. The comments below are really interesting and from serious mathematicians, so please have a look!
- An interesting inequality (sirjackshen.wordpress.com)
- Heuristic ideas about bounded prime gaps (motls.blogspot.com)
- Online reading seminar for Zhang’s “bounded gaps between primes” (terrytao.wordpress.com)