Integer partition formula

Some remarkable congruences for the partition function

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

In plain words, e.g., 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, .

Hardy-ramanujan formula

Srinivasa ramanujan contribution to mathematics

Ramanujan partition formula pdf

Ramanujan partition formula

Partition of 200

. .

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.

Background

In his 1919 paper,^[1] he proved the first two congruences using the following identities (using q-Pochhammer symbol notation):

He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these".

After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript em