HMMT 二月 2000 · 代数 · 第 10 题

HMMT February 2000 — Algebra — Problem 10

Find the smallest p ositiv e in teger a su h that x + a is not prime for an y in teger x .

解析

a = 1 learly do es not w ork, sin e if x = 1, then x + a = 2, whi h is prime. a = 2 4 2 learly do es not w ork, sin e if x = 1, then x + a = 5, whi h is also prime. Here is a table 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 2 4 2 2 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 in teger x . So, the answ er is a = 8 .