CONGRUENCES AND CONJECTURES FOR THE PARTITION FUNCTION

7

CoNJECTURE 5.1. (Subbarao [Su]) If A and B are integers with 0::::; B A,

then there are infinitely many integers n for which

p(An +B)

'f:

0 (mod 2).

This conjecture is known for every arithmetic progression B (mod A) for which

there is at least one n with p(An +B)

=

1 (mod 2) [Th. 2, 01]. The conjecture

is also known for every arithmetic progression B (mod A) where A is a power of 2

[Th. 1, B-0].

Unfortunately, very little is known about the partition function modulo 3. For

example, it is not even known that there are infinitely many n for which 3

I

p(n). As

an analogue to Conjecture 5.1, it seems reasonable to make the following conjecture.

CONJECTURE 5.2. If A and B are integers with 0

:S

B A, then there are

infinitely many integers n for which

p(An +B)

'f:

0 (mod 3).

Apart from Ramanujan's original congruences (2.2), (2.3), no others are known

where the modulus of the congruence equals the modulus of the arithmetic pro-

gression. In view of this and the work of Kiming and Olsson mentioned in the

introduction, we pose the following.

CONJECTURE 5.3. If£ ;:::: 13 is prime, and (3 is an integer, then there are

infinitely many integers n for which

p(Cn

+

(3)

'f:

0 (mod£).

Based on the results in

[A-0],

it seems reasonable to make the following conjecture.

CoNJECTURE 5.4. Suppose that£;:::: 5 is prime and that

p(An +B)

=

0 (mod £)

for every integer n. Then there exists (3

E

Sc such that {An+

B}

~

{

Cn

+

(3}.

We recall the following important conjecture of Newman

[N3].

CoNJECTURE 5.5. If M is a positive integer, then for every residue class r

(mod M) there are infinitely many integers n for which p(n)

=

r (mod M).

Although the results in

[A2,

01] provide a simple criterion for deducing Conjec-

ture 5.5 for any M coprime to 6, it remains open. In fact, Conjecture 5.5 has not

been proven for infinitely many M.

The remaining conjectures and problems are devoted to questions involving the

distribution of p(n) modulo integers M.

CoNJECTURE 5.6. If 0

:S

r M, then define

rSr(M,

X) by

r5r(M, X):= #{0

:S

n X :

~n)

=

r (mod M)}.

1. If 0

:S

r M, then there is a real number 0 dr ( M) 1 for which

lim r5r(M, X)

=

dr(M).

X--oo