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ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

  • MICHAEL D. HIRSCHHORN (a1) and JAMES A. SELLERS (a2)
Abstract
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).

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For correspondence; e-mail: sellersj@math.psu.edu
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[1] N. Baruah and B. Berndt , ‘Partition identities and Ramanujans modular equations’, J. Combin. Theory Ser. A 114(6) (2007), 10241045.

[2] J. Borwein and P. Borwein , ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323(2) (1991), 691701.

[3] J. Borwein , P. Borwein and F. Garvan , ‘Some cubic modular identities of Ramanujan’, Trans. Amer. Math. Soc. 343(1) (1994), 3547.

[4] A. Granville and K. Ono , ‘Defect zero p-blocks for finite simple groups’, Trans. Amer. Math. Soc. 348(1) (1996), 331347.

[6] M. Hirschhorn , F. Garvan and J. Borwein , ‘Cubic analogues of the Jacobian theta function θ(z,q)’, Canad. J. Math. 45(4) (1993), 673694.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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