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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.18742 |
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| _version_ | 1866913463656251392 |
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| author | Aragão, Lucas Chapman, Jonathan Ortega, Miquel Souza, Victor |
| author_facet | Aragão, Lucas Chapman, Jonathan Ortega, Miquel Souza, Victor |
| contents | The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2, \dotsc, N\}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2} \log N$ monochromatic solutions, and that the leading constant is sharp. For $r=3$ and $r=4$ we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the number of monochromatic solutions to multiplicative equations Aragão, Lucas Chapman, Jonathan Ortega, Miquel Souza, Victor Combinatorics Number Theory The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2, \dotsc, N\}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2} \log N$ monochromatic solutions, and that the leading constant is sharp. For $r=3$ and $r=4$ we obtain tight results up to a multiplicative logarithmic factor. We also provide bounds for more colours and other multiplicative equations. |
| title | On the number of monochromatic solutions to multiplicative equations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2311.18742 |