2

SCOTT AHLGREN AND KEN ONO

Ramanujan conjectured (and in some cases proved) extensions of these congruences

to powers of 5, 7, and 11. In fact, after his work

[Be-0, Rl, R2, R3]

and

subsequent work of Watson

[Wa]

and Atkin

[Atl],

it is now known that if 24m= 1

(mod 5a7b1F), then we have

(2.3)

Since the work of Ramanujan, further examples of congruences involving primes

£:::; 31 have been found (see the works of

[At2, At-0, At-SwDl, At-SwD2, N4,

L-0, W]). The first systematic treatment of such congruences came only recently,

when the second author

[02]

proved that for any prime£;::: 5, there exist infinitely

many (non-nested) congruences of the form

(2.4)

p(An +B)=

0 (mod£) for all

n.

The first author extended the method to prove that if £ ;::: 5 is prime and m is a

positive integer, then there exist infinitely many congruences of the form

(2.5)

p(An +B)

=

0 (mod

£ffi)

for all

n.

In fact, it is shown that such congruences exist for any modulus

M

which is coprime

to 6.

The residue class

-8£

(mod£) has always played a distinguished role in the

theory. Indeed, all of Ramanujan's congruences and their extensions (2.3) lie within

the progressions

fn -

h

Further, all of the congruences (2.4) and (2.5) whose

existence is proven in

[A2, 02]

necessarily lie within this progression. To further

highlight the importance of this class, we mention work of Kiming and Olsson

[K-0],

who proved that if£;::: 5 is prime and

p(fn

+

(3)

=

0 (mod£) for all

n,

then

f3

=

-8£

(mod£).

However, some of the examples alluded to above-in particular those given in

[At2,

N4]-illustrate that congruences may indeed lie outside of this class. Atkin

[At2],

for example, proved that

p(17303n + 237)

=

0 (mod 13).

These examples call into question the true importance of the class

-8£

(mod£).

In a recent paper

[A-0],

the two authors have shown that congruences for

p(n)

are much more widespread than was previously known. In fact, they show that the

class

-8£

(mod£) is just one of (£ + 1)/2 residue classes modulo £ in which the

partition function has similar congruence properties.

To state the main result in

[A-OJ

requires some notation. For each prime£;::: 5,

define the integer

f£

E {

±1} by

(2.6)

and let

St

denote the set of(£+ 1)/2 integers

(2.7) St

:=

{(3 E

{0,

1, ... , £- 1}

Then we have