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In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The Indian mathematician Srinivasa Ramanujan discovered the following

$$p(5k+4)\equiv 0 \pmod 5$$
$$p(7k+5)\equiv 0 \pmod 7$$
$$p(11k+6)\equiv 0 \pmod {11}.$$

Background

In his 1919 paper (Ramanujan, 1919), he gave proof for the first two congruences using the following identities (using q-Pochhammer symbol notation):

$$\sum_{k=0}^\infty p(5k+4)q^k=5\frac{(q^5)_\infty^5}{(q)_\infty^6}$$
$$\sum_{k=0}^\infty p(7k+5)q^k=7\frac{(q^7)_\infty^3}{(q)_\infty^4}+49q\frac{(q^7)_\infty^7}{(q)_\infty^8}.$$

then stated that "It appears there are no equally simple properties for any moduli involving primes other than these".

After Ramanujan died in 1920, G. H. Hardy, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs Eisenstein series.

In 1944, Freeman Dyson defined the rank function and conjectured the existence of a crank function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan successfully found such a function, and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7 and 11.

Extending results of A. O. L. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give

$$p(4063467631k+30064597)\equiv 0\pmod{31}.$$

Later Ken Ono conjectured that the elusive crank also satisfies exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg in his 2005 paper Partition Congruences and the Andrews–Garvan–Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences Paper of the Year prize.

A conceptual explanation for Ramanujan's observation was finally discovered in January 2011  by considering the Hausdorff dimension of the following P function in the l-adic topology:

$$P_l(b;z) := \sum_{n=0}^\infty p\left(\frac{l^bn+1}{24}\right)q^{n/24}.$$

It is seen to have dimension 0 only in the cases where l = 5, 7 or 11 and since the partition function can be written as a linear combination of these functions this can be considered a formalization and proof of

Ramanujan's observation.

In 2001, R.L. Weaver gave an effective algorithm for finding congruences of the partition function, and tabulated 76,065 congruences. This was extended in 2012 by F. Johansson to 22,474,608,014 congruences, one large example being

$$p(28995244292486005245947069k + 28995221336976431135321047) \equiv 0 \pmod{29}.$$

Tau-function, for which there are other so-called Ramanujan congruences
Rank of a partition
Crank of a partition

References

S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147–153.
"Cozzarelli Prize". National Academy of Sciences. June 2014. Retrieved 2014-08-06.
Ken Ono, Amanda Folsom, & Zach Kent (2011). l-adic properties of the partition function American Institute of Mathematics.
Ken Ono & Jan Bruinier (2011). AN ALGEBRAIC FORMULA FOR THE PARTITION FUNCTION American Institute of Mathematics.
R.L. Weaver, New congruences for the partition function, The Ramanujan Journal, 5 (2001), 53-63. 

F. Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, LMS Journal of Computation and Mathematics 15 (2012), 341-359. 

Ono, Ken (2000). "Distribution of the partition function modulo m". Ann. Math. (2) 151 (1): 293–307. doi:10.2307/121118. Zbl 0984.11050.
Ono, Ken (2004). The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics 102. Providence, RI: American Mathematical Society. ISBN 0-8218-3368-5. Zbl 1119.11026.
Ramanujan, S. (1919). "Some properties of p(n), the number of partitions of n". Proc. Cambridge Philos. Soc. 19: 207–210. JFM 47.0885.01.