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

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