HMMT 二月 2000 · 团队赛 · 第 9 题
HMMT February 2000 — Team Round — Problem 9
题目详情
英文原题
- Find all p ositiv e primes of the form 4 x + 1, for x an integer.
解析
英文解析
- It suÆ es to onsider x 1, sin e 4( x ) + 1 = 4( x ) + 1, and 4(0) + 1 = 1 is not prime.
4 4 2 2 2 2 222
So, 4 x + 1 = (4 x + 4 x + 1) 4 x = (2 x + 1) (2 x ) = (2 x + 1 2 x )(2 x + 1 + 2 x ).
2 2 4
F or in tegers x , b oth 2 x 2 x + 1 and 2 x + 2 x + 1 are in tegers, so this fa tors 4 x + 1
2 2 2
unless 2 x 2 x + 1 = 1 or 2 x + 2 x + 1 = 1. Sin e x > 0, then 2 x + 2 x + 1 > 1, so
2 2 4 2
we m ust have 2 x 2 x + 1 = 1. 2 x 2 x + 1 = 1 is absurd (4 x + 1 ; 2 x + 2 x + 1 > 0,
2 4 x +1 2 24
so 2 x 2 x + 1 = > 0), so we solv e 2 x 2 x + 1 = 1, or 2 x 2 x = 0, so
2 x +2 x +12
x ( x 1) = 0, and x = 0 or x = 1. W e have already reje ted x = 0, so the only ase leftis x = 1, or 4(1) + 1 = 5 .4