Competition problem — Number theory

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Number theory Difficulty 4.8 AIME Prove it United States

Problem:

Find an $n$ such that $n! - (n-1)! + (n-2)! - (n-3)! + \cdots \pm 1!$ is prime. Be prepared to justify your answer for
$\left\{\begin{array}{c} n, \\ {\left[\frac{n+225}{10}\right],} \\ n \leq 25 \end{array}\right.$
points, where $[N]$ is the greatest integer less than $N$ .

Solution

Solution:

$3, 4, 5, 6, 7, 8, 10, 15, 19, 41$ (26 points), $59, 61$ (28 points), $105$ (33 points), $160$ (38 points) are the only ones less than or equal to $335$ . If anyone produces an answer larger than $335$ , then we ask for justification to call their bluff. It is not known whether or not there are infinitely many such $n$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.