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Congruence properties of Apéry numbers. (English) Zbl 0428.10008

In his proof of the irrationality of \(\zeta(3)\) Apéry introduced a sequence of numbers defined by the recurrence relation
\[ n^3 = (34n^3 - 51n^2 + 27n - 5)a_{n-1} - (n - 1)^3 a_{n-2} \]
with initial values \(a_0 = 1\), \(a_1= 5\). He showed that \(a_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+1}{k}^2\) so the \(a_n\) are integers, a fact which is not apparent from the recursion. The authors deduce some congruence properties of these integers, for example, \(a_n\equiv 1\pmod 2\), and \(a_{5n+1}\equiv a_{5n+3}\equiv 0 \pmod 5\) for all \(n\geq 0\), and \(a_p\equiv 5\pmod{p^2}\) for all primes \(p\).

MSC:

11B37 Recurrences
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