The residue of \(p(N)\) modulo small primes
type: Item
instance of: scholarly article
title: The residue of \(p(N)\) modulo small primes
zbMATH Open document ID: 0907.11035
DOI: 10.1023/A:1009705807315
author: Ken Ono
published in: The Ramanujan Journal
publication date: 1999-03-02T00:00:00Z
review text: Let \(l\) be a prime. A partition of \(n\) is called \(l\)-affine, \(l\)-square affine accordingly as each part is a power of \(l\), an even power of \(l\), respectively. Let \(C(t,n)\) denote the number of \(t\)-core partitions of \(n\). The author first proves that if \(n<l^{2s+2}\), then \(p(n)\) is congruent \(\pmod l\) to a sum of products \(\prod^s_{i=0} C(l,a_i)\), the sum taken over all \(l\)-square affine representations of \(n\), namely \[ n= \sum^s_{i=0} a_il^{2i}. \] He obtains more specific equivalent congruences in the cases \(l=3, 5, 7, 11\). One by-product is an alternate proof of the Ramanujan congruences \(\pmod{l^i}\), where \(l\in \{5, 7, 11\}\) and \(i\in \{1,2\}\).
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 1203269
zbMATH Keywords: \(q\)-series
zbMATH Keywords: \(t\)-core partitions
zbMATH Keywords: Ramanujan congruences
reviewed by: Neville Robbins
MaRDI profile type: Publication
full work available at URL: a:1009705807315
OpenAlex ID: W4250876721
description: scientific article; zbMATH DE number 1203269