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A CONJECTURE OF MERCA ON CONGRUENCES MODULO POWERS OF 2 FOR PARTITIONS INTO DISTINCT PARTS

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  27 March 2023

JULIA Q. D. DU  and
DAZHAO TANG Open the ORCID record for DAZHAO TANG [Opens in a new window]
JULIA Q. D. DU
Affiliation:
School of Mathematical Sciences, Hebei International Joint Research Center for Mathematics and Interdisciplinary Science, Hebei Normal University, Shijiazhuang 050024, PR China e-mail: qddu@hebtu.edu.cn
DAZHAO TANG*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, PR China
*
e-mail: dazhaotang@sina.com
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Abstract

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa 30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$. We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$. As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$. Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$.

Keywords

congruencesinternal congruencespartitionsdistinct partsmodular forms

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 26 - 36
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was partially supported by the National Natural Science Foundation of China (No. 12201177), the Natural Science Foundation of Hebei Province (No. A2021205018), the Science and Technology Research Project of Colleges and Universities of Hebei Province (No. BJK2023092), the Doctor Foundation of Hebei Normal University (No. L2021B02), the Program for Foreign Experts of Hebei Province and the Program for 100 Foreign Experts Plan of Hebei Province. The second author was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ–MSX0387), the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509) and the Doctoral Start-up Research Foundation (No. 21XLB038) of Chongqing Normal University.

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