Volume 291, 2001
Congruences and conjectures for the partition function
Scott Ahlgren and Ken Ono
The topic of congruences for the partition function
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
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
= 1 and
= 0 if
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,
and Mare integers. Such congruences were, of course, first discovered
by Ramanujan. Throughout the paper,
5 will denote a prime number, and
will denote the integer
With this notation, Ramanujan's famous congruences take the form
0 (mod£) if£= 5, 7, or 11.
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