An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0\equiv \tau (5m+5)(\mathrm{mod}5)$

Abstract

We present a proof of Ramanujan's congruences $$p(5n+4)\equiv 0(\mathrm{mod}5)\text{ and }\tau (5n+5)\equiv 0(\mathrm{mod}5).$$ The proof only requires a limiting case of Jacobi's triple product, a result that Ramanujan knew well, and a technique which Ramanujan used himself to compute values of $\tau (n)$.

4 pages