HMMT 二月 2005 · 冲刺赛 · 第 44 题

HMMT February 2005 — Guts Round — Problem 44

Write down a set S of positive integers, all greater than 1, whose product is P , such that for each x ∈ S , x is a proper divisor of ( P/x ) + 1. Your score is 2 n , where n = | S | .

解析

Write down a set S of positive integers, all greater than 1, such that for each x ∈ S , x is a proper divisor of ( P/x ) + 1, where P is the product of all the elements of S . Your score is 2 n , where n = | S | . Remark: This question was posed by Stefan Zn´ am in 1972. Solutions exist for | S | ≥ 5, but an explicit solution has been found only for | S | ≤ 13. The simplest solution is S = { 2 , 3 , 11 , 23 , 31 } . A solution for | S | = 7 is given by S = { 2 , 3 , 11 , 17 , 101 , 149 , 3109 } . For more information, see http://mathworld.wolfram.com/ZnamsProblem.html .