推 callmedance :感謝 04/24 17:28
※ 編輯: yueayase (36.236.229.213), 03/20/2015 21:21:55
※ 引述《callmedance (NightFury)》之銘言：
: 已知 a是大於18的正整數
: 且 (a-18)^(1/2) + (4a+12)^(1/2) 為正整數
: 求 a = ?
(a-18)^(1/2) + (4a+12)^(1/2) 正整數
=> (a-18)^(1/2) 和 (4a+12)^(1/2) 都是正整數
=> (a-18) 和 (4a+12) 都是完全平方數
2 2
令 a-18 = m , 4a+12 = n, 且設m, n > 0
2 2
=> n - 4m = 84
2 2
a = n - 3m - 66
2 2
n - 4m = (n+2m)(n-2m) = 84
=> 考慮84的正因數
n+2m n-2m n m a
84 1 85/2 83/4 => x
42 2 22 10 118 => v
28 3 31/2 25/4 => x
21 4 25/2 17/4 => x
14 6 10 2 22 => v
12 7 19/2 5/4 => x
所以 a = 118 or 22
※ 文章網址: http://www.ptt.cc/bbs/Math/M.1398330934.A.0B6.html