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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

ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + ⋯ = 1.2020569 … {\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.2020569\ldots }

cannot be written as a fraction p / q {\displaystyle p/q} where p and q are integers. The theorem is named after Roger Apéry.

The special values of the Riemann zeta function at even integers 2 n {\displaystyle 2n} ( n > 0 {\displaystyle n>0}) can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function’s values are in general rational or not at the odd integers 2 n + 1 {\displaystyle 2n+1} ( n > 1 {\displaystyle n>1}) (though they are conjectured to be irrational).