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ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  02 July 2009

MICHAEL D. HIRSCHHORN  and
JAMES A. SELLERS
MICHAEL D. HIRSCHHORN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia (email: m.hirschhorn@unsw.edu.au)
JAMES A. SELLERS
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: sellersj@math.psu.edu)
*
✉For correspondence; e-mail: m.hirschhorn@unsw.edu.au
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Abstract

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In a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

Keywords

partitionsregular partitionscongruencesJacobi’s triple product identity

MSC classification

Secondary: 11P83: Partitions; congruences and congruential restrictions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 58 - 63
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Calkin, N., Drake, N., James, K., Law, S., Lee, P., Penniston, D. and Radder, J., ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers 8 (2008), #A60.Google Scholar