Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T15:13:33.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Super congruence for the Apéry numbers

Published online by Cambridge University Press:  22 January 2016

Tsuneo Ishikawa
Tsuneo Ishikawa*
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe 657, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let, for any n ≥ 0,

.

R. Apéry’s proof of the irrationality of ζ(2) and ζ(3) made use of these numbers (see [10]). As a result, many properties of the Apéry numbers were found (see [l]-[9]). In particular, Beukers and Stienstra showed the interesting congruence (see [11, Theorem 13.1]).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 118 , June 1990, pp. 195 - 202
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Beukers, F., Irrationality of π2, periods of an elliptic curve and Γ1(5), Progress in Math., 31, pp. 47–66, Birkhäuser, 1983.Google Scholar
[2] Beukers, F., Arithmetical properties of Picard-Fuchs equation, Progress in Math., 51, pp. 33–38, Birkhäuser, 1984.Google Scholar
[3] Beukers, F., Some congruence for Apéry numbers, J. Number Theory, 21 (1985), 141–155.Google Scholar
[4] Beukers, F., Another congruence for Apéry numbers, J. Number Theory, 25 (1987), 201–210.Google Scholar
[5] Chowla, S., Cowles, J. and Cowles, M., Congruence properties of Apéry numbers, J. Number Theory, 12 (1980), 188–190.CrossRefGoogle Scholar
[6] Gessel, I., Some congruence for the Apéry numbers, J. Number Theory, 14 (1982), 362–368.CrossRefGoogle Scholar
[7] Hamme, L. Van, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference on p-adic analysis, Hengelhoef, Belgium, 1986.Google Scholar
[8] Ishikawa, T., On Beukers’ conjecture, Kobe J. Math., 6 (1989), 49–52.Google Scholar
[9] Mimura, Y., Congruence properties of Apéry numbers, J. Number Theory, 16 (1983), 138–146.Google Scholar
[10] Poorten, A. J. van der, A proof that Euler missed · · · Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer, 1 (1979), 195–203.CrossRefGoogle Scholar
[11] Stienstra, J. and Beukers, F., On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3 surface, Math. Ann., 271 (1985), 269–304.Google Scholar