notation denotes the product defined by

A short proof of this result can be obtained from the partition function generating function.

Ramanujan also discovered congruences modulo 7 and 11:

The first one comes from Ramanujan's identity

Since 5, 7, and 11 are consecutive primes, one might think that there would be an analogous congruence for the next prime 13, for some . However, there is no congruence of the form for any prime

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