Congruences like Ramanujan's for powers of the partition function
type: Item
instance of: scholarly article
title: Congruences like Ramanujan's for powers of the partition function
zbMATH Open document ID: 0765.11041
DOI: 10.1007/BF01197051
author: I. Kiming
author: Jørn B. Olsson
published in: Archiv der Mathematik
publication date: 1993-02-10T00:00:00Z
review text: Let \(p(n)\) be the partition function. The congruences of Ramanujan mentioned in the title of this paper are the three congruences \(p(5n+4)\equiv 0\pmod 5\), \(p(7n+5)\equiv 0\pmod 7\), \(p(11n+6)\equiv 0 \pmod {11}\), each being true for all nonnegative integers \(n\). For each integer \(r\), the function \(p_ r(n)\) is defined by means of the equation \(P(q)^ r=\sum_{n=0}^ \infty p_ r(n)q^ n\), where \(P(q)=\sum_{n=0}^ \infty p(n)q^ n=\prod_{n=1}^ \infty (1-q^ n)^{-1}\) is the generating function for \(p(n)\). Then \(p_ r(n)\) is an integer for all integers \(r\) and positive integers \(n\). If \(\pi\) is a prime \(\geq 5\), \(r\) is an integer, and \(a\in[0,1,\dots,\pi- 1]\), then it is said that there is a congruence for \(p\) at \((\pi,r,a)\) if \(p_ r(n\pi+a)\equiv 0 \pmod \pi\) for all nonnegative integers \(n\). A congruence for \(p\) at \((\pi,r,a)\) is called exceptional if \(r\not\in[\pi- 1,\pi-3]\). The theory of reductions of modular forms modulo \(\pi\) is used to prove that if there is an exceptional congruence for \(p\) at \((\pi,r,a)\) then \(r\) is odd and \(24a\equiv r \pmod \pi\). It is then shown that there is a congruence for \(p\) at \((\pi,\pi-1,a)\) if and only if \(24a+1\) is a quadratic nonresidue modulo \(\pi\), and there is a congruence for \(p\) at \((\pi,\pi-3,a)\) if and only if either \(8a+1\equiv 0 \pmod \pi\) or \(8a+1\) is a quadratic nonresidue modulo \(\pi\). The final theorem of this paper states that, for all nonnegative integers \(n\), \[ \begin{alignedat}{3} p_ 3(11n+7)&\equiv 0\pmod {11},&\quad p_ 5(11n+8)&\equiv 0\pmod {11},&\quad p_ 7(11n+9)&\equiv 0\pmod {11},\\ p_ 3(17n+15)&\equiv 0\pmod {17},&\quad p_ 9(17n+11)&\equiv 0\pmod {17},&\quad p_{13}(17n+14)&\equiv 0\pmod {17},\\ p_ 7(19n+9)&\equiv 0\pmod {19}, &\quad p_ 9(19n+17)&\equiv 0\pmod {19}, &\quad p_{13}(19n+14)&\equiv 0\pmod {19}.\end{alignedat} \] {}.
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 119864
zbMATH Keywords: Ramanujan congruences
zbMATH Keywords: partition function
zbMATH Keywords: generating function
zbMATH Keywords: exceptional congruence
reviewed by: Betty Garrison
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 119864