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2 results

  • GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES

  • Part of
  • BERNARD L. S. LIN, ANDREW Y. Z. WANG
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 90 / Issue 2 / October 2014
    Published online by Cambridge University Press:
    20 May 2014, pp. 204-212
    Print publication:
    October 2014
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  • Recently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed an analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keith’s conjecture and get a stronger result, which implies that all of Keith’s results on congruences modulo 3 are consequences of our result.

  • ELEMENTARY PROOFS OF VARIOUS FACTS ABOUT 3-CORES

  • Part of
  • MICHAEL D. HIRSCHHORN, JAMES A. SELLERS
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 79 / Issue 3 / June 2009
    Published online by Cambridge University Press:
    17 April 2009, pp. 507-512
    Print publication:
    June 2009
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  • 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).