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?
[Other articles in category /math] permanent link