Ramanujan and the Partition Function

The partition function p(n) counts the number of ways to write n as a sum of positive integers. p(4) = 5: we have 4, 3+1, 2+2, 2+1+1, 1+1+1+1. p(100) = 190,569,292. Ramanujan and Hardy found a stunning asymptotic formula for p(n), and Ramanujan alone discovered congruences: p(5k+4) is always divisible by 5, p(7k+5) by 7, p(11k+6) by 11. These Ramanujan congruences have no simple explanation — they remain one of the most mysterious and beautiful results in number theory, still generating research today.