General Family of Congruences Modulo Large Powers of 3 for Cubic Partition Pairs

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New Zealand Journal of Mathematics

Vol. 47, (2017), Pages 43-56


D. S. Gireesh

Department of Mathematics,

Dayananda Sagar College of Engineering,

Shavige Malleshwara Hills,

Kumaraswamy Layout,

Bengaluru-560 078, Karnataka,

India.

mailto:gireeshdap@gmail.com

M. S. Mahadeva Naika

Department of Mathematics,

Bangalore University,

Central College Campus,

Bengaluru-560 001, Karnataka,

India.

mailto:msmnaika@rediffmail.com




Abstract Let b(n) denote the number of cubic partition pairs of n. Recently, Chern proved two of Lin's conjectures on cubic partition pairs and he asked a question about general family of congruences modulo large powers of 3 for cubic partition pairs. In this paper, we give affirmative answer to Chern's question by finding (\alpha,\beta,\ell) such that

b(3^\alpha n+\ell)\equiv 0\pmod{3^\beta}.

In the process, we also give alternative proofs for Lin's conjectures.

Keywords Partitions; Cubic Partition Pairs; Congruences.

Classification (MSC2000) 05A17, 11P83.

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