HMMT 二月 2000 · 代数 · 第 10 题
HMMT February 2000 — Algebra — Problem 10
题目详情
英文原题
- Find the smallest p ositiv e integer a such that x + a is not prime for an y integer x .
解析
英文解析
- a = 1 learly do es not w ork, sin e if x = 1, then x + a = 2, which is prime. a = 2
4 2
learly do es not w ork, sin e if x = 1, then x + a = 5, which is also prime. Here is atable for a 's and v alues of x that sho w they do not w ork:
4 2
a x a + x
3 10 10009
4 1 17
5 2 41
6 1 37
7 20 160049
4 2 22
So, let us onsider a = 8 - i.e. the sum x + 64. This is the same as ( x + 8) 16 x =
2 2
( x + 4 x + 8)( x 4 x + 8 ) b y the di eren e of squares. This is learly not prime for an y integer x . So, the answ er is a = 8 .