Congruences for the partition function P include the classical congruences modulo 5, 7, and
11. The asymptotic density of odd
values of 
is conjectured to exist and equal 
(Judge et al. 2018), whereas odd
values of the partition function Q
have asymptotic density zero. Kolberg (1959) proved
that there are infinitely many even and odd
values of 
. In the congruences below,
is a nonnegative integer, 
is a positive integer,
and 
,
,
,
and 
are positive integers.
Leibniz noted that 
is prime for 
, 3, 4, 5, 6, but not 7. In fact, values of 
for which 
is prime are 2, 3, 4, 5,
6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063),
corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575).
Numbers which cannot be written as a product of 
are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064),
corresponding to numbers of nonisomorphic Abelian
groups which are not possible for any group order.
Ramanujan (1919) conjectured a number of congruences involving 
. In particular, he proved
 |
(1)
|
using Ramanujan's identity (Darling 1919, Hardy and Wright 1979, Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that
 |
(2)
|
and Krečmar (1933) proved that
 |
(3)
|
Watson (1938) then proved the general congruence
 |
(4)
|
(Gordon and Hughes 1981; Hardy 1999, p. 89). For 
, 2, ..., the corresponding minimal values of 
are 4, 24, 99, 599, 2474, 14974, 61849, ... (OEIS A052462).
However, the even more general congruences
 |
(5)
|
 |
(6)
|
seem also to hold for 
, 99, 124 and 
, 2474, 3099, respectively.
Ramanujan (1919) showed that
 |
(7)
|
(Darling 1919), which can be derived using the Euler identity and Jacobi triple product (Hardy 1999, pp. 87-88), and also that
 |
(8)
|
(Hardy 1999, p. 90). He conjectured that in general
![P(n)=0 (mod 7^b) if 24n=1 (mod 7^b) [incorrect]](/images/equations/PartitionFunctionPCongruences/NumberedEquation9.svg) |
(9)
|
(Gordon and Hughes 1981, Hardy 1999), although Gupta (1936) showed that this is false when 
.
For example, 
but 
. Watson (1938) subsequently formulated
and proved the modified relation
 |
(10)
|
for 
.
The minimal nonnegative integers solving the
congruence on 
for 
, 2, ... are 0, 47, 2301, 112747, ..., where the initial
0 for 
corresponds to modulus 1 and is not an instance of Watson's
congruence (OEIS A052463).
However, the even more general congruences
 |
(11)
|
appear to hold for 
, 33, 40, 47.
Ramanujan (1919) showed that
 |
(12)
|
holds (Gordon and Hughes 1981; Hardy 1999, pp. 87-88), and conjectured the general relation
 |
(13)
|
This was finally proved by Atkin (1967). For 
, 2, ..., the corresponding minimal values of 
are 6, 116, 721, 14031, ... (OEIS A052465).
Atkin and O'Brien (1967) proved
 |
(14)
|
where 
is an integer depending only on 
(Gordon and Hughes 1981). For 
, 2, ..., the corresponding minimal values of 
are 6, 162, 1007, 27371, ... (OEIS A052466).
Subbarao (1966) conjectured that in every arithmetic progression 
(mod 
), there are infinitely many integers 
for which 
is even, and infinitely
many integers 
for which 
is odd. Radu (2012) proved
the odd part of this conjecture, completing its proof
together with the even case proved by Ono (1996).
Dyson (1944) introduced the rank of a partition, its largest part minus its number of parts, to explain the congruences
modulo 5 and 7. Since this statistic does not explain the congruence
modulo 11, he conjectured the existence of a different statistic, the crank. Andrews
and Garvan (1988) found the partition crank and
used it to explain all three congruences. Mahlburg
(2005) proved a conjecture of Ono that the partition
crank satisfies general congruences analogous
to those for 
.
In the Season 4 opening episode "Trust Metric" (2007) of the television crime drama NUMB3RS, math genius Charlie Eppes closes the opening scene by informing his class they will cover partition congruences in the next class (despite the strange fact that the current lesson was on Nash equilibria).
and 
Modulo Powers of 13." Trans. Amer. Math. Soc. 126,
442-459, 1967.Chowla, S. "Congruence Properties of Partitions."
J. London Math. Soc. 9, 247, 1934.Darling, H. B. C.
"On Mr. Ramanujan's Congruence Properties of 
." Proc. Cambridge Philos. Soc. 19, 217-218,
1919.Darling, H. B. C. "Proofs of Certain Identities
and Congruences Enunciated by S. Ramanujan." Proc. London Math. Soc. 19,
350-372, 1921.Drost, J. L. "A Shorter Proof of the Ramanujan
Congruence mod 5." Amer. Math. Monthly 104, 963-964, 1997.Dyson,
F. J. "Some Guesses in the Theory of Partitions." Eureka 8,
10-15, 1944.Dyson, F. J. "Mappings and Symmetries of Partitions."
J. Combin. Theory Ser. A 51, 169-180, 1989.Folsom, A.;
Kent, Z. A.; and Ono, K. "
-Adic Properties of the Partition Function." 
." In 
."
Ann. Math. 151, 293-307, 2000.Radu, C.-S. "A Proof
of Subbarao's Conjecture." J. reine. angew. Math. 672, 161-175,
2012. 
, the Number of Partitions of 
." Proc. Cambridge Philos. Soc. 19, 207-210,
1919.Ramanujan, S. "Congruence Properties of Partitions."
Math. Z. 9, 147-153, 1921.Sloane, N. J. A. Sequences
." J. Combin. Th. 6, 56-59,
1969.