IJPAM: Volume 104, No. 4 (2015)

SOME INTERESTING RESULTS ON $7$-CORE PARTITIONS

Kuwali Das
Department of Mathematical Sciences
Bodoland University
Kokrajhar, 783370, Assam, INDIA


Abstract. If $p\geq 5$ is a prime with $-7$ is a quadratic non-residue modulo $p$ then for any non-negative integers $n,$ $7$-core partitions of $n$ satisfy several interesting results, for example

\begin{displaymath} a_{7}(\left( 49\cdot p^{2k}n+ 7(r+1)p^{2k}-2\right ) = 49 a_{7}(\left( 7\cdot p^{2k}n+ (r+1)p^{2k}-2\right ) \end{displaymath}

where $r\in\{ 2,4,5\}$ and $a_7(n)$ denotes the number of $7$-cores of $n$.

Received: June 16, 2015

AMS Subject Classification:

Key Words and Phrases: $t$-core partition, theta function, dissection, congruence

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DOI: 10.12732/ijpam.v104i4.7 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 104
Issue: 4
Pages: 551 - 559


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