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.


11B37 Recurrences
11A07 Congruences; primitive roots; residue systems
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.