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

TL;DR

This paper proves that the largest prime divisor of expressions like n!+1 grows at least proportionally to n, improving previous bounds through new combinatorial techniques.

Contribution

It establishes a stronger lower bound on the growth of the largest prime divisor of n!+f(n), extending previous results with novel combinatorial methods.

Findings

 limsup_{n   + P(n!+1)/n   7.238.

 limsup_{n   + P(n!+f(n))/n   7.238.

The new combinatorial idea improves bounds on prime divisors of factorial-related expressions.

Abstract

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.