Bulletin of the Australian Mathematical Society

Research Article

ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

MICHAEL D. HIRSCHHORNa1 and JAMES A. SELLERSa2 c1

a1 School of Mathematics and Statistics, UNSW, Sydney 2052, Australia (email: m.hirschhorn@unsw.edu.au)

a2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (email: sellersj@math.psu.edu)

Abstract

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

(Received September 18 2008)

2000 Mathematics subject classification

  • primary 05A17;
  • 11P81;
  • 11P83

Keywords and phrases

  • partition;
  • 3-cores;
  • generating function;
  • Jacobi’s Triple Product Identity;
  • congruences;
  • Lambert series

Correspondence:

c1 For correspondence; e-mail: sellersj@math.psu.edu