PUMaC 2019 · 代数（B 组） · 第 7 题

PUMaC 2019 — Algebra (Division B) — Problem 7

Let ω = e and ζ = e . Let S = { ( a, b ) ∈ Z | 0 ≤ a ≤ 2016 , 0 ≤ b ≤ 2018 , ( a, b ) 6 = (0 , 0) } . ∏ a b Compute ( ω − ζ ). ( a,b ) ∈ S

解析

Let ω = e and ζ = e . Let S = { ( a, b ) ∈ Z | 0 ≤ a ≤ 2016 , 0 ≤ b ≤ 2018 , ( a, b ) 6 = (0 , 0) } . ∏ a b Compute ( ω − ζ ). ( a,b ) ∈ S Answer: 4072323 Proposed by: Frank Lu 2018 2018 ∏ ∏ b 2019 a b 2019 a First, fix a . Note that ( x − ζ ) = x − 1. Hence, if a 6 = 0, ( ω − ζ ) = ω − 1. For b =0 b =0 2018 2018 2018 2018 ∏ ∏ ∏ ∑ b b b b a = 0, we have that this is (1 − ζ ) = 2019, since ( x − ζ ) = ( x − ζ ) / ( x − 1) = x . b =1 b =1 b =0 b =0 2016 ∏ 2019 a Thus, our product becomes ( ω − 1) ∗ 2019. But note that this then becomes 2017 ∗ 2019, a =1 2 2019 a since the ω are just a permutation of the 2017th roots of unity besides 1 (as 2017 and 2019 are relatively prime), which is then just 4072323 .