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ON A CURIOUS PROPERTY OF BELL NUMBERS

Part of: Combinatorics Elementary number theory Sequences and sets

Published online by Cambridge University Press:  14 June 2011

ZHI-WEI SUN  and
DON ZAGIER
ZHI-WEI SUN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China (email: zwsun@nju.edu.cn)
DON ZAGIER
Affiliation:
Max-Planck-Institut für Mathematik, 53111 Bonn, Germany Collège de France, 75005 Paris, France (email: don.zagier@mpim-bonn.mpg.de)
*
✉For correspondence; e-mail: zwsun@nju.edu.cn
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Abstract

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In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.

Keywords

Bell numbersderangement numberscongruences modulo a prime

MSC classification

Secondary: 11B73: Bell and Stirling numbers 05A18: Partitions of sets 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 153 - 158
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first author is supported by the National Natural Science Foundation (grant 10871087) and the Overseas Cooperation Fund (grant 10928101) of China.

References

[1]Sun, Z. W., ‘A conjecture on Bell numbers’ (a message to Number Theory List), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1007&L=nmbrthry&T=0&P=1066.Google Scholar
[2]Touchard, J., ‘Propriétés arithmétiques de certains nombres recurrents’, Ann. Soc. Sci. Bruxelles 53A (1933), 21–31.Google Scholar