Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915251505594368 |
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| author | Luca, Florian Noubissie, Armand |
| author_facet | Luca, Florian Noubissie, Armand |
| contents | Here, we show that if $u_n=n2^n\pm 1$, then the largest prime factor of $u_n\pm m!$ for $n\ge 0,~m\ge 2$ tends to infinity with $\max\{m,n\}$. In particular, the largest $n$ participating in the equation $u_n\pm m!=2^a3^b5^c7^d$ with $n\ge 1,~m\ge 2$ is $n=8$ for which $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root Luca, Florian Noubissie, Armand Number Theory B.6.5; D.6.1 Here, we show that if $u_n=n2^n\pm 1$, then the largest prime factor of $u_n\pm m!$ for $n\ge 0,~m\ge 2$ tends to infinity with $\max\{m,n\}$. In particular, the largest $n$ participating in the equation $u_n\pm m!=2^a3^b5^c7^d$ with $n\ge 1,~m\ge 2$ is $n=8$ for which $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$. |
| title | Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root |
| topic | Number Theory B.6.5; D.6.1 |
| url | https://arxiv.org/abs/2504.14513 |