PUMaC 2010 · 代数（B 组） · 第 3 题
PUMaC 2010 — Algebra (Division B) — Problem 3
题目详情
- Write √ = a + b 2 + c 4 + d 8 + e 16, with a , b , c , d , and e integers. Find a + b + 5 2 − 1 2 2 2 c + d + e . x 4
解析
- Write = a + b 2 + c 4 + d 8 + e 16, with a , b , c , d , and e integral. Find a + b + 5 2 − 1 2 2 2 c + d + e . √ √ 5 5 1 / 5 Solution: 5. By multiplying both sides by 2 − 1 and noting that the numbers 1, 2 = 2 , √ √ √ 5 5 5 2 / 5 3 / 5 4 / 5 4 = 2 , 8 = 2 , and 16 = 2 are all linearly independent over Q , we can set up five equations for five unknowns, whose solution is a = b = c = d = e = 1. x 4