Contemporary Mathematics

Volume 291, 2001

Congruences and conjectures for the partition function

Scott Ahlgren and Ken Ono

1. Introduction

The topic of congruences for the partition function

p(

n) has been widely stud-

ied. The purpose of this paper is threefold. In the first part we give an account

of some of

the contributions which the two authors have made to the area in the

past several years. In the second part we present a new construction of certain

modular forms related to the partition function; this gives a new (and particu-

larly simple) framework in which to consider congruences for the partition function

modulo primes

f.

2:

5. Finally, we will pose some conjectures which we hope will

clarify some of the interesting remaining questions on the congruential distribution

of values of p(n).

2. Recent results

A partition of a positive integer n is a non-increasing sequence of positive in-

tegers whose sum is n. Let p( n) denote the number of partitions of n (we define

p(O)

= 1 and

p(a)

= 0 if

a

tf.

Z;::::

0

).

We are concerned with the topic of linear

congruences for the partition function; i.e. relations of the form

p(An +B)= 0 (mod M) for all n,

where

A, B,

and Mare integers. Such congruences were, of course, first discovered

by Ramanujan. Throughout the paper,

f.

2:

5 will denote a prime number, and

8e

will denote the integer

(2.1)

With this notation, Ramanujan's famous congruences take the form

(2.2)

p(fn-

8e)

=

0 (mod£) if£= 5, 7, or 11.

1991

Mathematics Subject Classification.

Primary 11P83; Secondary 05A17.

Key words and phrases.

Ramanujan-type congruences for the partition function.

Both authors thank the National Science Foundation for its generous support. The second

author also thanks the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation

for their generous support.

©

2001 American Mathematical Society

http://dx.doi.org/10.1090/conm/291/04889