On the main partition page, three partition congruences first stated by Ramanujan were listed: \begin{align*} p(5k+4)&\equiv 0 (\mod 5), \,\,\forall k\geq 0,\\ p(7k+5)&\equiv 0 (\mod 7), \,\,\forall k\geq 0,\\ p(11k+6)&\equiv 0 (\mod 11), \,\,\forall k\geq 0. \end{align*} Ramanujan conjectured three more general congruences, namely, that for each integer $m\geq 1$ and each integer, \begin{align*} p(5^mk+\delta_5(m))&\equiv 0 (\mod 5^m), \,\,\forall k\geq 0,\\ p(7^mk+\delta_7(m))&\equiv 0 (\mod 7^m), \,\,\forall k\geq 0,\\ p(11^mk+\delta_{11}(m))&\equiv 0 (\mod 11^m), \,\,\forall k\geq 0, \end{align*} where, for $l \in \{5,7,11\}$, $0< \delta_l(m) < l^m$ and and $24\delta_l(m)\equiv 1 (\mod l^m)$. The second congruence was only partially correct, and for $m>2$ the correct version is \[ p(7^m k+\delta_7(m))\equiv 0 (\mod 7^{\lfloor m/2 \rfloor +1}), \,\,\forall k\geq 0. \] Various proofs of these congruences were given, including those in the papers [At1], [R1], [R2] and [W1].

Congruence for other primes, although not so simple, were stated by others, including the authors in [At2] and [AB1], who gave \begin{align*} p(59^4 · 13n + 111247) &\equiv 0 (\mod 13), \\ p(23^3 · 17n + 2623) &\equiv 0 (\mod 17). \end{align*}

This collection of congruences was greatly generalized by Ahlgren and Ono. In [O], Ono showed that if $M\geq 5$ is prime, then there are infinitely many pairs of positive integers $A$ and $B$ for which \[ p(An+B)\equiv 0(\mod M) \] for every non-negative integer $n$. Ahlgren in [Ah] extended this result to all positive integers $M$ that are coprime to 6. These results were further extended by Ahlgren and Ono in their joint paper [AhO].

Congruence results are known for other partition functions, and examples of these may be found on the pages for colored partitions or multipartitions, generalized Frobenius partitions, k-regular partitions, overpartitions, plane partitions, smallest-parts functions for partitions and t-core partitions.

[Ah] S. Ahlgren, Distribution of the partition function modulo composite integers $M$. Math. Ann. 318 (2000), no. 4, 795–803.

[AhO] S. Ahlgren and K. Ono, Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 12882–12884.

[At1] A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), pages 14-32.

[At2] A. O. L. Atkin, Congruence Hecke operators, Proc. Symp. Pure Math. 12 (1969), 33–40.

[AB1] A. O. L. Atkin and J. N. O’Brien, Some properties of $p(n)$ and $c(n)$ modulo powers of 13, Trans. Amer. Math. Soc. 126 (1967), 442–459.

[O] K. Ono, Distribution of the partition function modulo $m$. Ann. of Math. (2) 151 (2000), no. 1, 293–307.

[R1] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2) 19, pages 207-210.

[R2] S. Ramanujan, Ramanujan’s unpublished manuscript on the partition and tau-functions (Commentary by B. C. Berndt and K. Ono) The Andrews Festschrift (Ed. D. Foata and G.-N. Han), Springer-Verlag, Berlin, 2001, pages 39-110.

[W1] G. N. Watson, Ramanujan’s vermutung über zerfällungsanzahlen, J. reine Angew. Math. 179 (1938), pages 97-128.

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