A new proof of the Ramanujan congruences for the partition function

Abstract

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

$$\begin{array}{rcl}p(5n+4)&\equiv&0\pmod{5},\\[6pt]p(7n+5)&\equiv&0\pmod{7},\\[6pt]p(11n+6)&\equiv&0\pmod{11}.\end{array}$$

These congruences have been proven via q-series identities, combinatorial arguments, and the theory of Hecke operators. We present a new proof which does not rely on any specialized identities or combinatorial constructions, nor does it necessitate introducing Hecke operators. Instead, our proof follows from simple congruences between the coefficients of modular forms, basic properties of Klein’s modular j-function, and the chain rule for differentiation. Furthermore, this proof naturally encompasses all three congruences in a single argument.