HMMT 二月 2001 · 冲刺赛 · 第 60 题

HMMT February 2001 — Guts Round — Problem 60

[ ∞ ] Find an n such that n ! − ( n − 1)! + ( n − 2)! − ( n − 3)! + · · · ± 1! is prime. Be { n, n ≤ 25 prepared to justify your answer for points, where [ N ] is the greatest integer n +225 , n> 25 [ ] 10 less than N .

解析

[ ∞ ] Find an n such that n ! − ( n − 1)! + ( n − 2)! − ( n − 3)! + · · · ± 1! is prime. Be { n, n ≤ 25 prepared to justify your answer for points, where [ N ] is the greatest integer n +225 , n> 25 [ ] 10 less than N . 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 . 9