## 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

### Keywords:

congruence properties; Apery numbers
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