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Congruence properties for the partition function

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Congruence properties for the partitionfunction †‡ Scott Ahlgren* and Ken Ono *Department of Mathematics, University of Illinois, Urbana, IL 61801; andDepartment of Mathematics, University of Wisconsin, Madison, WI 53706 Communicated by Richard A. Askey, University of Wisconsin, Madison, WI, September 17, 2001 (received for review August 15, 2000) Eighty years ago, Ramanujan conjectured and proved some strikingand provide a theoretical framework that (to our knowledge) congruences for the partition function modulo powers of 5, 7, andexplains every known congruence for the partition function. 11. Until recently, only a handful of further such congruences wereFor each prime5, define the integer{1} by known. Here we report that such congruences are much more 6 widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ra:,[1.3] manujantype congruence. and letSdenote the set of (1)2 integers 1. Introduction and Statement of Results.    Letp(n) denote the usual partition function;p(n) is the S:0, 1, . . . ,1:0 or. number of ways to write a positive integernas the sum of a [1.4] nonincreasing sequence of positive integers. As usual, we agree thatp(0)1 and thatp(t)0 ift0. Many of the most THEOREM1.If5is prime, m is a positive integer, andS, interesting arithmetic properties of this function were suggested then a positive proportion of the primes Q 1 (mod24)have (and often proved) by Ramanujan. Notice that ifis defined by the property that 2 13 Q n1 24:,[1.1] m p0mod24 then the celebrated Ramanujan congruences may be written for all n124(mod24)withgcd(Q,n)1. succinctly in the form Note that the case when (mod) already contains the main results in refs. 13 and 17. pn0mod. In general, there is no simple description of the set of primes Qoccurring inTheorem 1.However, as Atkin (12) showed, when Countless papers have been written on these three congruences 5, 7, or 13, the situation can be made quite explicit. For and their extensions (already conjectured, and in some cases example, Atkin proved the following (see ref. 12 for analogous proved, by Ramanujan) to arbitrary powers of 5, 7, and 11 [see results when7 or 13). the fundamental works of Andrews, Atkin, Dyson, Garvan, Kim, THEOREM2 [Atkin (12)]. Ramanujan, Stanton, and SwinnertonDyer (1–10)]. Each of these extensions lies within the class(mod(1)). The importantSuppose4 (mod5)is prime and n is a positive integer with role that this class plays in the theory is illustrated by the work n.If n23(mod120)or n47(mod120),then of Kiming and Olsson (ref. 11, theorem 1), who proved that if3 n1 5 is prime andp(n)0 (mod) for alln, then (mod).24p0mod 5. Work of Atkin, Newman, O’Brien, and SwinnertonDyer (12, m(2)Suppose3 (mod5)is a prime exceeding 3, and nis a 14, 15, 16) produced further congruences modulofor primes positive integer with (n) 1.If n23 (mod120)or 31 and smallm. The examples discovered by Atkin and n47 (mod120),then Newman in refs. 12 and 16 show that not every congruence lies within the progression(mod). For example, we have 2 n1 24p0mod 5. p17303n2370mod 13.[1.2] We should remark that Newman (16) discovered the simplest We have shown (13, 17) that if5 is prime andmis any example of the congruences described in the first part of positive integer, then there are infinitely many congruences of Theorem 2(i.e., the case where19). Notice that in either part the form ofTheorem 2, fixingnin an appropriate residue class modulo m pAnB0mod. 120yields a Ramanujantype congruence. For example, if13, then the second part ofTheorem 2implies, for every integer As in the case of Ramanujan’s congruences, all of these arith n, that metic progressions lie within the class(mod). To summa p10985n26970mod 5.[1.5] rize, the current state of knowledge consists of a systematic theory of congruences within the progressions(mod), as Arguing in this manner fromTheorem 1, we obtain well as some sporadic examples of congruences that fall outside THEOREM3.If5is prime, m is a positive integer, andof this class. In view of this, it is natural to wonder what role the S,then there are infinitely many nonnested arithmetic progres class(mod) truly plays. sions{AnB}{n},such that for every integer n we have In this paper, we show that in general this class is not as m distinguished as might have been expected. In fact, we prove that pAnB0mod. it is only one of (1)2 classes moduloin which the partition function enjoys similar congruence properties. The results in this paper include the main results in refs. 13 and 17 as special casesTo whom reprint requests should be addressed. Email: ono@math.wisc.edu.
P SNovember 6, 2001vol. 98no. 23 12882–12884NA
www.pnas.orgcgidoi10.1073pnas.191488598