Another congruence for the Apéry numbers. (English) Zbl 0614.10011

In 1979 Apéry introduced the numbers \[ a_ n=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)^ 2 \left( \begin{matrix} n+k\\ k\end{matrix} \right)^ 2 \] in his irrationality proofs for \(\zeta\) (3). Subsequently various writers discovered interesting congruence properties of these numbers (there are papers in J. Number Theory 12, 14, 16 and 21; see the citations of the present paper or Zbl 0425.10033; Zbl 0428.10008; Zbl 0482.10003; Zbl 0504.10007; Zbl 0571.10008). The author notes that the generating function for the \(a_ n\) satisfies a certain differential equation which is the symmetric square of a second order linear differential equation and that the function can be interpreted as a period of a family of K3 surfaces [see the author and C. A. M. Peters, J. Reine Angew. Math. 351, 42-54 (1984; Zbl 0541.14007)].
In the present note the author uses an ad hoc method to give a proof for a new congruence for the \(a_ n\) by relating the generating function for the \(a_ n\) to a certain modular form; however he remarks that the congruence must arise from the interplay between the numbers \(a_ n\) and the \(\zeta\)-function of a certain algebraic threefold and it is likely that a stronger congruence holds.


11B37 Recurrences
11F33 Congruences for modular and \(p\)-adic modular forms
11A07 Congruences; primitive roots; residue systems
14J30 \(3\)-folds
11F11 Holomorphic modular forms of integral weight
Full Text: DOI


[1] Beukers, F., Some congruences for the apéry numbers, J. number theory, 21, 141-155, (1985) · Zbl 0571.10008
[2] Beukers, F., Irrationality proofs using modular forms, Journées arithmétiques besançon, (1985), Astérisque, to appear · Zbl 0613.10031
[3] Beauville, A., LES familles stables de courbes elliptiques sur P2 admettant quatre fibres singulières, C. R. acad. sci. Paris, 294, 657, (1982) · Zbl 0504.14016
[4] Beukers, F.; Peters, C.A.M., A family of K3 surfaces and ζ(3), J. reine angew. math., 351, 42-54, (1984) · Zbl 0541.14007
[5] Chowla, S.; Cowles, J.; Cowles, M., Congruence properties of apéry numbers, J. number theory, 12, 188-190, (1980) · Zbl 0428.10008
[6] Dwork, B., On apéry’s differential operator, “groupe d’étude d’analyse ultramétrique”, Paris, (1980/1981)
[7] Fano, G., Uber lineare homogene differentialgleichungen, Math. anal., 53, 493-590, (1900) · JFM 31.0342.01
[8] Gessel, I., Some congruences for the apéry numbers, J. number theory, 14, 362-368, (1982) · Zbl 0482.10003
[9] Koike, M., On Mckay’s conjecture, Nagoya math. J., 95, 85-89, (1984) · Zbl 0548.10018
[10] Ogg, A., ()
[11] {\scC. A. M. Peters}, Monodromy and Picard-Fuchs equations for families of K3-surfaces and elliptic curves, preprint. · Zbl 0612.14006
[12] 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
[13] Rademacher, H., ()
[14] Stienstra, J.; Beukers, F., On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3 surfaces, Math. ann., 271, 269-304, (1985) · Zbl 0539.14006
[15] Shimura, G., Introduction to the arithmetic theory of automorphic forms, (1971), Iwanami Shoten Princeton
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.