2 results
INFINITE FAMILIES OF CONGRUENCES MODULO 3 AND 9 FOR BIPARTITIONS WITH 3-CORES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 91 / Issue 1 / February 2015
- Published online by Cambridge University Press:
- 27 August 2014, pp. 47-52
- Print publication:
- February 2015
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Let
$A_{3}(n)$ denote the number of bipartitions of 
$n$ with 3-cores. Recently, Lin [‘Some results on bipartitions with 3-core’, J. Number Theory139 (2014), 44–52] established some congruences modulo 4, 5, 7 and 8 for 
$A_{3}(n)$. In this paper, we prove several infinite families of congruences modulo 3 and 9 for 
$A_{3}(n)$ by employing two identities due to Ramanujan.
INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 2 / April 2013
- Published online by Cambridge University Press:
- 07 June 2012, pp. 304-315
- Print publication:
- April 2013
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Let u(n) and v(n) be the number of representations of a nonnegative integer n in the forms x2+4y2+4z2 and x2+2y2+2z2, respectively, with x,y,z∈ℤ, and let a4(n) and r3(n) be the number of 4-cores of n and the number of representations of n as a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we prove that
$u(8n+5)=8a_4(n)=v(8n+5)=\frac {1}{3}r_3(8n+5)$. With the help of this and a classical result of Gauss, we find a simple proof of a result on a4 (n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’, Acta Arith. 80 (1997), 249–272]. We also find some new infinite families of arithmetic relations involving a4 (n) .
