PUMaC 2013 · 数论（A 组） · 第 7 题

PUMaC 2013 — Number Theory (Division A) — Problem 7

[ 7 ] Suppose P ( x ) is a degree n monic polynomial with integer coefficients such that 2013 divides P ( r ) for exactly 1000 values of r between 1 and 2013 inclusive. Find the minimum value of n . 11 16

解析

[ 7 ] Suppose P ( x ) is a degree n monic polynomial with integer coefficients such that 2013 divides P ( r ) for exactly 1000 values of r between 1 and 2013 inclusive. Find the minimum value of n . Solution Let a, b, and c be the number of solutions to the congruence equations P ( x ) ≡ 0 (mod 3) , P ( x ) ≡ 0 (mod 11) , P ( x ) ≡ 0 (mod 61) respectively. Then abc = 1000 and a ≤ min { 3 , n } , b ≤ min { 11 , n } , c ≤ min { 61 , n } . Then max a = 2 and max b = 10 so that min c = 50. In particular, min n = 50. The existence of such P is clear. 2 11 16