ON THE LARGEST PRIME DIVISOR OF <i>n</i> ! + 1

Abstract For an integer $m&gt;1$ , we denote by $P(m)$ the largest prime divisor of m . We improve a result of Stewart [‘On the greatest and least prime factors of ${n!}+1$ , II’, Publ. Math. Debrecen 65 (3–4) (2004), 461–480] by showing that $\limsup _{n \rightarrow \infty } P({n!}+1)/n \geqslant 1+9\log 2$ . More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup _{n \rightarrow \infty } P({n!}+f(n))/n \geqslant 1+9\log 2$ . This improves a result of Luca and Shparlinski [‘Prime divisors of shifted factorials’, Bull. Lond. Math. Soc. 37 (6) (2005), 809–817]. These improvements stem from an additional combinatorial idea that builds upon the works mentioned above.