An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$
Hartosh Singh Bal, Gaurav Bhatnagar
公開日: 2025/8/30
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 $\tau(n)$.