Congruences via Partitions with Exactly Two Part Sizes
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Jirattikansakul, Sittinon Kittipassorn, Teeradej Kongsiri, Kraiwich Moonwichit, Nitipon Sriamorn, Kirati |
| author_facet | Jirattikansakul, Sittinon Kittipassorn, Teeradej Kongsiri, Kraiwich Moonwichit, Nitipon Sriamorn, Kirati |
| contents | We prove the congruence $\sum_{1 \leq k < \sqrt{N}} σ_0 (N - k^2) \equiv 0 \pmod 4$, where $σ_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114) \}$. Our proof relies on a result of Keith which states that $ν_2 (N) \equiv 0 \pmod 4$, where $ν_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25394 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Congruences via Partitions with Exactly Two Part Sizes Jirattikansakul, Sittinon Kittipassorn, Teeradej Kongsiri, Kraiwich Moonwichit, Nitipon Sriamorn, Kirati Number Theory Combinatorics 11P83 (Primary) 05A17 (Secondary) We prove the congruence $\sum_{1 \leq k < \sqrt{N}} σ_0 (N - k^2) \equiv 0 \pmod 4$, where $σ_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114) \}$. Our proof relies on a result of Keith which states that $ν_2 (N) \equiv 0 \pmod 4$, where $ν_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques. |
| title | Congruences via Partitions with Exactly Two Part Sizes |
| topic | Number Theory Combinatorics 11P83 (Primary) 05A17 (Secondary) |
| url | https://arxiv.org/abs/2604.25394 |