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| Format: | Preprint |
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2013
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| Online Access: | https://arxiv.org/abs/1302.5708 |
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| _version_ | 1866909212899016704 |
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| author | Sellers, James A. |
| author_facet | Sellers, James A. |
| contents | In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors.
In 2011, Baruah and Sarmah proved a number of congruence properties for $cϕ_4$, all with moduli which are powers of 4. In this brief note, we add to the collection of congruences for $cϕ_4$ by proving this function satisfies an unexpected result modulo 5. The proof is elementary, relying on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1302_5708 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions Sellers, James A. Number Theory 05A17, 11P83 In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. In 2011, Baruah and Sarmah proved a number of congruence properties for $cϕ_4$, all with moduli which are powers of 4. In this brief note, we add to the collection of congruences for $cϕ_4$ by proving this function satisfies an unexpected result modulo 5. The proof is elementary, relying on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan. |
| title | An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions |
| topic | Number Theory 05A17, 11P83 |
| url | https://arxiv.org/abs/1302.5708 |