HMMT 二月 2016 · 代数 · 第 2 题

英文解析

2. For which integers n ∈ { 1 , 2 , . . . , 15 } is n + 1 a prime number?
   Proposed by: Evan Chen
   Answer: 1,2,4
   n = 1 works. If n has an odd prime factor, you can factor, and this is simulated also by n = 8:
   2 k
   ∑
   2 k +1 ia + 1 = ( a + 1)( ( − a ) )
   i =0
   with both parts larger than one when a > 1 and k > 0. So it remains to check 2 and 4, which work.
   Thus the answers are 1,2,4.