Solvability of $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913391875981312 |
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| author | Allen, Meaghan |
| author_facet | Allen, Meaghan |
| contents | Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ if and only if $x=a=1$. We do this by looking at different cases depending on the values of $x$ and $k$. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, Stănică, Shanta, Shorey and Nair are also used. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solvability of $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ Allen, Meaghan Number Theory 11B65 (Primary) 05A10 (Secondary) Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ if and only if $x=a=1$. We do this by looking at different cases depending on the values of $x$ and $k$. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, Stănică, Shanta, Shorey and Nair are also used. |
| title | Solvability of $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ |
| topic | Number Theory 11B65 (Primary) 05A10 (Secondary) |
| url | https://arxiv.org/abs/2406.10404 |