In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number :\zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}=\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\ldots = 1.2020569\ldots cannot be written as a fraction p/q with p and q being integers. History Euler proved in the eighteenth century that if n is a positive integer then we have :\frac{1}{1^{2n}}+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\ldots = \frac{p}{q}\pi^{2n} for some rational number p/q. Specifically, writing the infinite series on the left as ζ(2n) he...