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$23$-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  23 December 2022

DAVID PENNISTON
DAVID PENNISTON*
Affiliation:
Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI 54901-8631, USA
*
e-mail: [email protected]
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Abstract

A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$.

Keywords

partitionscongruencesmodular forms

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 254 - 263
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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