回覆 11# Harris 的帖子
第 15 題
試問滿足\(m^2-4n\)及\(n^2-4m\)皆為完全平方數的正整數解對\((m,n)\)共有幾組?
(A)1 (B)2 (C)3 (D)4 (E)5
[解答]
m^2 - 4n 和 n^2 - 4m 都是完全平方數
(1) 當 m < 5,僅有 (m,n) = (4,4) 一解
(2) 當 m ≧ 5
m^2 - 4m - (m - 3)^2 = 2m - 9 > 0
m^2 - 4m > (m - 3)^2
不失一般性,設 m ≧ n
m^2 > m^2 - 4n ≧ m^2 - 4m > (m - 3)^2
(i) m^2 - 4n = (m - 1)^2
2m - 4n = 2(m - 2n) = 1,不合
(ii) m^2 - 4n = (m - 2)^2
m - n = 1
n^2 - 4m = n^2 - 4(n + 1) = (n - 2)^2 - 8 = t^2
(n + t - 2)(n - t - 2) = 8
n = 5,m = 6
故有 (m,n) = (6,5)、(5,6)、(4,4) 三解