# The Universe of Discourse

Thu, 31 Jan 2008

Ramanujan's congruences
Let p(n) be the number of partitions of the integer n. For example, p(4) = 5 because there are 5 partitions of the integer 4, namely {4, 3+1, 2+2, 2+1+1, 1+1+1+1}.

Ramanujan's congruences state that:

 p(5k+4) =0 (mod 5) p(7k+5) =0 (mod 7) p(11k+6) =0 (mod 11)

Looking at this, anyone could conjecture that p(13k+7) = 0 (mod 13), but it isn't so; p(7) = 15 and p(20) = 48·13+3.

But there are other such congruences. For example, according to Partition Congruences and the Andrews-Garvan-Dyson Crank:

$$p(17\cdot41^4k + 1122838) = 0 \pmod{17}$$

Isn't mathematics awesome?