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

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
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