Character sums and products of factorials modulo $p$

Moubariz Z. Garaev; Florian Luca

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Displaying similar documents to “Character sums and products of factorials modulo $p$ ”

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On the largest prime factor of $n! + 2^{n} - 1$

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For an integer $n \geq 2$ we denote by $P (n)$ the largest prime factor of $n$ . We obtain several upper bounds on the number of solutions of congruences of the form $n! + 2^{n} - 1 \equiv 0 (mod q)$ and use these bounds to show that $\underset{n \to \infty}{lim sup} P (n! + 2^{n} - 1) / n \geq (2 π^{2} + 3) / 18 .$