Arithmetic properties of generalized Frobenius partitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916950111682560 |
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| author | Ahlgren, Scott Castillo, Cruz |
| author_facet | Ahlgren, Scott Castillo, Cruz |
| contents | Ramanujan proved three famous congruences for the partition function modulo 5, 7, and 11. The first author and Boylan proved that these congruences are the only ones of this type. In 1984 Andrews introduced the $m$-colored Frobenius partition functions $cϕ_m$; these are natural higher-level analogues of the partition function which have attracted a great deal of attention in the ensuing decades. For each $m\in \{5, 7, 11\}$ there are two analogues of Ramanujan's congruences for $cϕ_m$, and for these $m$ we prove there are no congruences like Ramanujan's other than these six. Our methods involve a blend of theory and computation with modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties of generalized Frobenius partitions Ahlgren, Scott Castillo, Cruz Number Theory 11P83, 11F33 Ramanujan proved three famous congruences for the partition function modulo 5, 7, and 11. The first author and Boylan proved that these congruences are the only ones of this type. In 1984 Andrews introduced the $m$-colored Frobenius partition functions $cϕ_m$; these are natural higher-level analogues of the partition function which have attracted a great deal of attention in the ensuing decades. For each $m\in \{5, 7, 11\}$ there are two analogues of Ramanujan's congruences for $cϕ_m$, and for these $m$ we prove there are no congruences like Ramanujan's other than these six. Our methods involve a blend of theory and computation with modular forms. |
| title | Arithmetic properties of generalized Frobenius partitions |
| topic | Number Theory 11P83, 11F33 |
| url | https://arxiv.org/abs/2509.12078 |