Husonghu，寒潭清，Jenny，和野菜花在一起比较他们的银行存款数。发现每两个人的银行存款数加起来是一个完全平方数，四个人的银行存款数加起来是也一个完全平方数。每个人分别有多少钱？
注：每个人的银行存款数都是整数，四个人的钱数各不相同，而且每个人至少有100块钱。
野菜花已经解的很好了，我只是加一点关于通解的想法。设这四个数的和为n^2。则n^2可以用3种不同的形式表成两个数的平方和。根据Fermat定理的推广（见平面整点问题），n一定满足下两个条件之一：n有两个形如4k+1的素因子，或n有一个形如4k+1的素因子的三次方。找到这样的数后，再用野菜花的方法算出四个数来。有时算出负数，要加以排除，有时是分数，要再乘一个倍数。
这样的素数包括5，13，17，29，...，满足条件的最小的数是65，就是野菜花发现的。（你的Excel怎么什么都能算，我的为什么不能？）下一个是85。算出来的四个数是590，4594，5810，17906。
I used excel to find the three pairs whose sum of squres are the same perfect square in the following way: I enter 15 in C1 and A3, and enter =A3+1 in A4, then drag the small square on the bottom right corner down, you get first column:15,16,17,... Enter =A3^2 in B3, then drag the small square down, you get :225,256,289,...for the 2nd column. Enter =C1+1 in D1, then drag the sall square to the right, you get: 15,16,17,... for the 1st row Enter =C1^2 in C2, then drag the small square to the right , get 225,256,289,... for the second row. Enter =(B3-225)^.5 in C3, and drag the small square down, get the 3rd column. Enter =(B3-256)^.5 in D3, and drag the small square down, get the 4th column. etc If there is an integer which is very easy to see, then there is a pair of integers whose sum of squares is a perfect square.
Don't be mad if I said too much in details, maybe some other people didn't know. |
Very creative use of Excel. The disigners at Microsoft will never know how mathematicians use their software. |
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