  For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$.
We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log
2>7.238$, which improves a result of Stewart. More generally, for any nonzero
polynomial $f(X)$ with integer coefficients, we show that $\limsup_{n
\rightarrow +\infty} P(n!+f(n))/n \geqslant 1+9\log2$. This improves a result
of Luca and Shparlinski. These improvements come from an additional
combinatoric idea to the works mentioned above.
