An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914015100272640 |
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| author | Bal, Hartosh Singh Bhatnagar, Gaurav |
| author_facet | Bal, Hartosh Singh Bhatnagar, Gaurav |
| contents | We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } τ(5n+5) \equiv 0 \pmod 5.$$ 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 $τ(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$ Bal, Hartosh Singh Bhatnagar, Gaurav Number Theory Combinatorics 11P83, 05A17 We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } τ(5n+5) \equiv 0 \pmod 5.$$ 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 $τ(n)$. |
| title | An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv τ(5m+5) \pmod 5$ |
| topic | Number Theory Combinatorics 11P83, 05A17 |
| url | https://arxiv.org/abs/2509.00532 |