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PROOF OF SOME CONJECTURAL CONGRUENCES OF DA SILVA AND SELLERS

Part of: Additive number theory; partitions Combinatorics Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 December 2021

AJIT SINGH Open the ORCID record for AJIT SINGH [Opens in a new window]  and
RUPAM BARMAN Open the ORCID record for RUPAM BARMAN [Opens in a new window]
AJIT SINGH
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Assam 781039, India e-mail: ajit18@iitg.ac.in
RUPAM BARMAN*
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Assam 781039, India
*
e-mail: rupam@iitg.ac.in
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Abstract

Let $p_{\{3, 3\}}(n)$ denote the number of $3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc. 104(3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences using the theory of modular forms.

Keywords

partition3-regularthree coloursmodular forms

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 57 - 61
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

da Silva, R. and Sellers, J. A., ‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc. 104(3) (2021), 415–423.Google Scholar
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Radu, S., ‘An algorithmic approach to Ramanujan’s congruences’, Ramanujan J. 20(2) (2009), 295–302.CrossRefGoogle Scholar
Radu, S. and Sellers, J. A., ‘Congruence properties modulo $5$ and $7$ for the pod function’, Int. J. Number Theory 7(8) (2011), 2249–2259.CrossRefGoogle Scholar
Wang, L., ‘Arithmetic properties of $\left(k,\ell \right)$ -regular bipartitions’, Bull. Aust. Math. Soc. 95 (2017), 353–364.CrossRefGoogle Scholar