A polynomial method for proving congruences of pω(n,m) and pν(n,m)
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Published:2023-02-06
Issue:
Volume:
Page:1-9
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ISSN:1793-0421
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Container-title:International Journal of Number Theory
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language:en
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Short-container-title:Int. J. Number Theory
Author:
Xia Ernest X. W.1,
Zhao Xiang2
Affiliation:
1. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu Province, P. R. China
2. School of Mathematical Sciences, Jiangsu University, Jiangsu, Zhenjiang 212013, P. R. China
Abstract
Recently, two partition functions [Formula: see text] and [Formula: see text] were introduced by Andrews et al. The partition function [Formula: see text] denotes the number of partitions of [Formula: see text] in which each odd part is less than twice the smallest part, and [Formula: see text] counts the number of partitions of [Formula: see text] into distinct non-negative parts such that all odd parts are less than twice the smallest part. Very recently, Silva et al. studied congruence properties of the restricted partition functions [Formula: see text] and [Formula: see text] which denote the number of partitions enumerated by [Formula: see text] and [Formula: see text], respectively, into exactly [Formula: see text] parts. In this paper, we give a polynomial method for discovering congruences for [Formula: see text] and [Formula: see text] by checking a finite number of initial values. Employing the polynomial method, we proved new and existing congruences for [Formula: see text] and [Formula: see text] based on a bounded number of calculations. For example, in order to prove that [Formula: see text] holds for all [Formula: see text], it suffices to show that the congruence holds when [Formula: see text].
Funder
National Natural Science Foundation of China
Natural Science Foundation of Jiangsu Province of China
Publisher
World Scientific Pub Co Pte Ltd
Subject
Algebra and Number Theory