Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915433974595584 |
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| author | Sellers, James A. |
| author_facet | Sellers, James A. |
| contents | In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all $n\geq 0$, \begin{align*} a(4n+2) &\equiv 0 \pmod{2}, \textrm{\ \ and} \\ a(4n+3) &\equiv 0 \pmod{2}. \end{align*} Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice Sellers, James A. Number Theory 11P83, 05A17 In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all $n\geq 0$, \begin{align*} a(4n+2) &\equiv 0 \pmod{2}, \textrm{\ \ and} \\ a(4n+3) &\equiv 0 \pmod{2}. \end{align*} Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences. |
| title | Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice |
| topic | Number Theory 11P83, 05A17 |
| url | https://arxiv.org/abs/2409.12321 |