: Mathématiques: Apéry's theorem

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Apéry's theorem

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
${\\displaystyle \\zeta (3)=\\sum _{n=1}^{\\infty }{\ rac {1}{n^{3}}}={\ rac {1}{1^{3}}}+{\ rac {1}{2^{3}}}+{\ rac {1}{3^{3}}}+\\cdots =1.2020569\\ldots }$
cannot be written as a fraction ${\\displaystyle p/q}$ where p and q are integers. The theorem is named after Roger Apéry. The special values of the Riemann zeta function at even integers ${\\displaystyle 2n}$ ( ${\\displaystyle n>0}$ ) 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 ${\\displaystyle 2n+1}$ ( ${\\displaystyle n>1}$ ) (though they are conjectured to be irrational).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_theorem)