# Apéry numbers

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The Apéry numbers $a _ { n }$, $b _ { n }$ are defined by the finite sums
for every integer $n \geq 0$. They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of $\zeta ( 3 )$ and $\zeta ( 2 ) = \pi ^ { 2 } / 6$, respectively. In the case of $\zeta ( 3 )$, Apéry showed that there exists a sequence of rational numbers $c _ { n }$ with denominator dividing $\operatorname { lcm } ( 1 , \ldots , n ) ^ { 3 }$ such that $0 < | a _ { n } \zeta ( 3 ) - c _ { n } | < ( \sqrt { 2 } - 1 ) ^ { 4 n }$ for all $n > 0$. Together with the fact that $\operatorname { lcm } ( 1 , \dots , n ) > 3 ^ { n }$, this implies the irrationality of $\zeta ( 3 )$. For a very lively and amusing account of Apéry's discovery, see [a4]. In 1979 F. Beukers [a1] gave a very short irrationality proof of $\zeta ( 3 )$, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of $\zeta ( 5 )$ so far (2001).
T. Rival [a5] proved the very surprising result that $\zeta ( 2 n + 1 ) \notin \mathbf{Q}$ for infinitely many $n$.
It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, $a _ { m p ^ r} \equiv a _ { m p ^ { r - 1 } } ( \operatorname { mod } p ^ { 3 r } )$ for all positive integers $m$, $r$ and all prime numbers $p \geq 5$. Another congruence is $a _ {( p - 1 )/ 2 } \equiv \gamma _ { p } ( \operatorname { mod } p )$ for all prime numbers $p \geq 5$. Here, $\gamma _ { n }$ denotes the coefficient of $q ^ { n }$ in the $q$-expansion of a modular cusp form. For more details see [a2], [a3].