# How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ?

I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?

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You might want to review The Irrationality of $\zeta(3)$. Apery's Proof is on the Wiki for comparison to the other link. Here is another source for Apery's proof. Regards. –  Amzoti Jan 18 '13 at 23:49
I don't know which recursion you mean but recursion can indeed give rise to irrationality. A classic example is $x_{n+1}=(x_n+2/x_n)/2$, which converges to $\sqrt 2$ for any $x_0>0$. –  lhf Jan 19 '13 at 0:42
I meant the recursion Apery used (which is more complicated imho). –  mick Jan 21 '13 at 16:27
For an overview see this answer to the question Apéry's constant (ζ(3)) value. –  Américo Tavares Jan 21 at 1:31
@mick Are you familiar with the fact that if a number can be approximated through rationals arbitrarily good (omitting 'how much good' as it won't really fit in the comment) then it is likely to be a transcendental? This is called diophantine approximation, you might like to get some references on a brief review of them. –  Balarka Sen Feb 8 at 20:14