Séminaire Lotharingien de Combinatoire, B93a (2025), 4 pp.
Hartosh Singh Bal and Gaurav Bhatnagar
An Easy Proof of
Ramanujan's Famous Congruences p(5m+4) ≡ 0 ≡
τ(5m+5) (mod 5)
Abstract.
We present a proof of Ramanujan's congruences
p(5n+4) ≡ 0 (mod 5) and τ(5n+5)
≡ 0 (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 τ(n).
Received: August 4, 2025.
Revised: August 26, 2025.
Accepted: August 26, 2025.
The following versions are available: