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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2411.13146 |
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| _version_ | 1866912320053051392 |
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| author | Lambard, Guillaume |
| author_facet | Lambard, Guillaume |
| contents | The Erdös-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $, and for $ k \geq 2 $ it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erdös-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods Lambard, Guillaume Number Theory Combinatorics The Erdös-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $, and for $ k \geq 2 $ it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erdös-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture. |
| title | An Analytical Exploration of the Erdös-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2411.13146 |