Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T06:18:37.513Z Has data issue: false hasContentIssue false
  • English
  • Français

A simple proof of Watson's partition congruences for powers of 7

Published online by Cambridge University Press:  09 April 2009

F. G. Garvan
F. G. Garvan
Affiliation:
School of Mathematics, University of New South WalesPost Office Box 1 Kensington, N.S.W. 2033, Australia The Pennsylvania State UniversityDepartment of Mathematics University Park, Pennsylvania 16802, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

Ramanujan conjectured that if n is of a specific form then p(n), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson.

In this paper we establish appropriate generating formulae, from which Watson's results follow easily.

Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 3 , June 1984 , pp. 316 - 334
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Atkin, A. O. L., ‘Ramanujan congruences for p −k(n)’, Canad. J. Math. 20 (1968), 67–78.CrossRefGoogle Scholar
[2]Chowla, S., ‘Congruence properties of partitions’, J. London Math. Soc. 9 (1934), 247.CrossRefGoogle Scholar
[3]Gupta, H., ‘A table of partitions’, Proc. London Math. Soc. (2) 39 (1935), 142–149.CrossRefGoogle Scholar
[4]Hirschhorn, M. D. and Hunt, D. C., ‘A simple proof of the Ramanujan conjecture for powers of 5’, J. Reine Angew. Math. 326 (1981), 1–17.Google Scholar
[5]Kolberg, O., ‘Some identities involving the partition function’, Math. Scand. 5 (1957), 77–92.CrossRefGoogle Scholar
[6]Ramanujan, S., ‘Some properties of p(n), the number of partitions of n’, Proc. Cambridge Philos. Soc. 19 (1919), 207–210.Google Scholar
[7]Watson, G. N., ‘Ramanujans Vermutung über Zerfällungsanzahlen’, J. Reine Angew. Math. 179 (1938), 97–128.CrossRefGoogle Scholar
[8]Zuckerman, H. S., ‘Identities analogous to Ramanujan's identities involving the partition function’, Duke Math. J. 5 (1939), 88–119.CrossRefGoogle Scholar