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HMMT 二月 2017 · 团队赛 · 第 8 题

HMMT February 2017 — Team Round — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 45 ] Does there exist an irrational number α > 1 such that n b α c ≡ 0 (mod 2017) for all integers n ≥ 1?
解析
  1. [ 45 ] Does there exist an irrational number α > 1 such that n b α c ≡ 0 (mod 2017) for all integers n ≥ 1? Proposed by: Sam Korsky Answer: Yes 2 n n n Let α > 1 and 0 < β < 1 be the roots of x − 4035 x + 2017. Then note that b α c = α + β − 1. Let n n x = α + β for all nonnegative integers n . It’s easy to verify that x = 4035 x − 2017 x ≡ x n n n − 1 n − 2 n − 1 (mod 2017) so since x = 4035 ≡ 1 (mod 2017) we have that x ≡ 1 (mod 2017) for all n . Thus α 1 n satisfies the problem.