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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2408.01306 |
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| _version_ | 1866910895738716160 |
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| author | Chan, Tsz Ho |
| author_facet | Chan, Tsz Ho |
| contents | In this article, we are interested in whether a product of three consecutive integers $a (a+1) (a+2)$ divides another such product $b (b+1) (b+2)$. If this happens, we prove that there is some gaps between them, namely $b \gg \frac{a \log a)^{1/6}}{\log \log a)^{1/3}}$. We also consider other polynomial sequences such as $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$ for some fixed integer $l$. Our method is based on effective Liouville-Baker-Feldman theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Diophantine equation $b (b+1) (b+2) = t a (a + 1) (a + 2)$ and gap principle Chan, Tsz Ho Number Theory 11D25 In this article, we are interested in whether a product of three consecutive integers $a (a+1) (a+2)$ divides another such product $b (b+1) (b+2)$. If this happens, we prove that there is some gaps between them, namely $b \gg \frac{a \log a)^{1/6}}{\log \log a)^{1/3}}$. We also consider other polynomial sequences such as $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$ for some fixed integer $l$. Our method is based on effective Liouville-Baker-Feldman theorem. |
| title | The Diophantine equation $b (b+1) (b+2) = t a (a + 1) (a + 2)$ and gap principle |
| topic | Number Theory 11D25 |
| url | https://arxiv.org/abs/2408.01306 |