HMMT 二月 2016 · 代数 · 第 2 题
HMMT February 2016 — Algebra — Problem 2
题目详情
- For which integers n ∈ { 1 , 2 , . . . , 15 } is n + 1 a prime number?
解析
- 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 i a + 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.