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Infinite families of congruences modulo 5 and 7 for the cubic partition function

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  16 January 2019

Olivia X. M. Yao
Olivia X. M. Yao*
Affiliation:
Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013P. R. China (yaoxiangmei@163.com)
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Abstract

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In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

Keywords

cubic partitionscongruencesθ function identities

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1189 - 1205
Copyright
Copyright © Royal Society of Edinburgh 2019 

References

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