Ramanujan's congruences

In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan:

${\displaystyle {\begin{aligned}p(5k+4)&\equiv 0{\pmod {5}},\\p(7k+5)&\equiv 0{\pmod {7}},\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}}$

In plain words, the first congruence means that if a number is 4 more than a multiple of 5, i.e. it is in the sequence

4, 9, 14, 19, 24, 29, . . .

then the number of its partitions is a multiple of 5.

Later, other congruences of this type were discovered, for numbers and for Tau-functions.