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arXiv · 2509.00532

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

Hartosh Singh Bal·Gaurav Bhatnagar
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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 $\tau(n)$.

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Hartosh Singh Bal, Gaurav Bhatnagar. 2025-08-30. An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv \tau(5m+5) \pmod 5$. https://arxiv.org/abs/2509.00532

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