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Some congruences for the Apéry numbers. (English) Zbl 0571.10008

In 1978 R. Apéry introduced the numbers \(a_ n=\sum^{n}_{0}\left( \begin{matrix} n\\ k\end{matrix} \right)^ 2 \left( \begin{matrix} n+k\\ k\end{matrix} \right)\) and \(u_ n=\sum^{n}_{0}\left( \begin{matrix} n\\ k\end{matrix} \right)^ 2 \left( \begin{matrix} n+k\\ k\end{matrix} \right)^ 2\) in his irrationality proofs for \(\zeta\) (2), \(\zeta\) (3). Generalizing previous results, it is shown that for any prime p and any m,r\(\in {\mathbb{N}}\) one has \(a_{mp^ r-1}\equiv a_{mp^{r-1}-1}\quad (mod p^{3r}).\) The same congruences hold for \(u_ n\). The proofs are tedious but straightforward.

MSC:

11B37 Recurrences
11A07 Congruences; primitive roots; residue systems
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References:

[1] Beukers, F, Irrationality of π2, periods of an elliptic curve and γ1 (5), (), 47-66
[2] Beukers, F; Peters, C, A family of K3-surfaces and ζ(3), J. reine angew. math., 351, 42-54, (1984) · Zbl 0541.14007
[3] Stienstra, J; Beukers, F, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. annalen, 271, 269-304, (1985) · Zbl 0539.14006
[4] Chowla, S; Cowles, J; Cowles, M, Congruence properties of apéry numbers, J. number theory, 12, 188-190, (1980) · Zbl 0428.10008
[5] Dwork, B, Norm residue symbol in local number fields, Abh. math. sem. univ. Hamburg, 22, 180-190, (1958) · Zbl 0083.26001
[6] Gessel, I, Some congruences for apéry numbers, J. number theory, 14, 362-368, (1982) · Zbl 0482.10003
[7] Hazewinkel, M, ()
[8] Mimura, Y, Congruence properties of apéry numbers, J. number theory, 16, 138-146, (1983) · Zbl 0504.10007
[9] van der Poorten, A.J, A proof that Euler missed…apéry’s proof of the irrationality of ζ(3), Math. intelligencer, 1, 195-203, (1979) · Zbl 0409.10028
[10] Radoux, C, Quelques propriétés arithmétiques des nombres d’apéry, C. R. acad. sci. Paris, A291, 567-569, (1980) · Zbl 0445.10013
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