Arithmetic properties of 7-regular partitions

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Abstract

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),
$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$
We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

Keywords

Partitions Congruences 7-Regular partitions Modular equation 

Mathematics Subject Classification

Primary 05A17 Secondary 11P83 

Notes

Acknowledgements

The author thanks the referee for his/her careful reading of the manuscript and helpful suggestions which improved the presentation of this work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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