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arXiv · 2504.14513

Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root

Florian Luca·Armand Noubissie
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Abstract

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 $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$.

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Florian Luca, Armand Noubissie. 2025-04-20. Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root. https://arxiv.org/abs/2504.14513

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