The Partition Congruences

What Ramanujan Discovered

Ramanujan discovered striking congruence properties satisfied by the partition function $p(n)$. For every nonnegative integer $n$, the number of partitions of $5n+4$ is divisible by 5, the number of partitions of $7n+5$ is divisible by 7, and the number of partitions of $11n+6$ is divisible by 11.

Understanding $p(n)$

The partition function $p(n)$ counts the number of ways of writing $n$ as a sum of positive integers, where order does not matter. For example, $p(4)=5$, corresponding to the five partitions:

Why These Congruences Matter

Partition congruences revealed deep arithmetic structure within a function defined by pure combinatorics. They connect partition theory to modular forms and number theory, showing that $p(n)$ carries hidden regularities when viewed modulo small integers.

Proof and Later Developments

Ramanujan announced proofs in his original notebook. Later mathematicians proved these congruences by studying the generating function of $p(n)$, a modular form of weight $$\begin{gathered} - \\ frac12 \end{gathered}$$. This led to further congruences, including those modulo 13, 17, 19, and beyond.

Original Source

S. Ramanujan, Notebook entry dated 1919. Published in Proceedings of the Cambridge Philosophical Society, Vol. 19 (1919), pp. 207-210.

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