On the largest prime factor of $n!+{2}^{n}-1$
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 859-870.

For an integer $n\ge 2$ we denote by $P\left(n\right)$ the largest prime factor of $n$. We obtain several upper bounds on the number of solutions of congruences of the form $n!+{2}^{n}-1\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ and use these bounds to show that

 $\underset{n\to \infty }{lim sup}P\left(n!+{2}^{n}-1\right)/n\ge \left(2{\pi }^{2}+3\right)/18.$

Pour un entier $n\ge 2$, notons $P\left(n\right)$ le plus grand facteur premier de $n$. Nous obtenons des majorations sur le nombre de solutions de congruences de la forme $n!+{2}^{n}-1\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ et nous utilisons ces bornes pour montrer que

 $\underset{n\to \infty }{lim sup}P\left(n!+{2}^{n}-1\right)/n\ge \left(2{\pi }^{2}+3\right)/18.$

DOI: 10.5802/jtnb.524
Florian Luca 1; Igor E. Shparlinski 2

1 Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
2 Macquarie University Sydney, NSW 2109, Australia
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Florian Luca; Igor E. Shparlinski. On the largest prime factor of $n!+ 2^n-1$. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 859-870. doi : 10.5802/jtnb.524. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.524/

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