The overpartition function modulo small powers of 2
type: Item
instance of: scholarly article
title: The overpartition function modulo small powers of 2
zbMATH Open document ID: 1114.11083
author: Karl Mahlburg
published in: Discrete Mathematics
publication date: 2004-11-22T00:00:00Z
review text: The main result of this note is that \(\overline{p}(n) \equiv 0 \pmod{64}\) for almost all natural numbers \(n\), where \(\overline{p}(n)\) denotes the number of overpartitions of \(n\). This result is obtained by relating \(\overline{p}(n)\) modulo \(64\) to a linear combination of the functions \(c_k(n)\), for \(1 \leq k \leq 5\), and then studying the \(2\)-adic properties of these functions. Here \(c_k(n)\) denotes the number of representations \(n = n_1^2 + \cdots + n_k^2\), where each \(n_i\) is positive. The author notes that preliminary calculations suggest that the main result may be true modulo arbitrary powers of \(2\). However, the methods of this paper ``run into serious limitations'' for powers of \(2\) beyond \(64\).
reviewed by: Jeremy Lovejoy
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 2117319
zbMATH Keywords: overpartitions
zbMATH Keywords: congruences
zbMATH Keywords: sums of squares
MaRDI profile type: Publication
full work available at URL: j.disc.2004.03.014
OpenAlex ID: W2008843962
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DOI: 10.1016/J.DISC.2004.03.014
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description: scientific article; zbMATH DE number 2117319