On the largest prime divisor of $n!+1$

Abstract

For an integer $m>1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that ${\mathrm{limsup}}_{n\to +\infty }P(n!+1)/n⩾1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that ${\mathrm{limsup}}_{n\to +\infty }P(n!+f(n))/n⩾1+9\log 2$. This improves a result of Luca and Shparlinski. These improvements come from an additional combinatoric idea to the works mentioned above.

11 pages