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".
Forty years later, George Andrews and Frank Garvan found such a function, and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7 and 11.
For example: Extending the results of A. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6.
For example, his results give Later Ken Ono conjectured that the elusive crank also satisfies exactly the same types of general congruences.
[2] A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 [3] by considering the Hausdorff dimension of the following