# An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv \tau(5m+5) \pmod 5$

## TL;DR

This paper provides a simplified proof of Ramanujan's well-known congruences for the partition function and Ramanujan's tau function modulo 5, utilizing classical identities and Ramanujan's own techniques.

## Contribution

It offers an accessible proof of Ramanujan's congruences using basic identities and historical methods, avoiding complex modern machinery.

## Key findings

- Proof of $p(5n+4) \\equiv 0 \\pmod 5$
- Proof of $\tau(5n+5) \\equiv 0 \\pmod 5$
- Simplifies understanding of Ramanujan's congruences

## Abstract

We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } \tau(5n+5) \equiv 0 \pmod 5.$$ 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 $\tau(n)$.