欢迎光临散文网 会员登陆 & 注册

1988.IMO.6数论--无穷递降法

2023-08-25 16:08 作者:封包  | 我要投稿

        我们的情人不过是随便借个名字，用幻想吹出来的肥皂泡，把信拿去吧，你可以使假戏成真。我本来是无病呻吟，漫无目的的吐露爱情，现在这些漂泊不定的鸟儿有地方栖息了，你可以从信里看出来。拿去吧，由于不是出自真心，话就说得格外动听，拿去吧，就这么办！

——《西哈诺》

%7B%5CLARGE%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A(IMO.1988.6)%E5%B7%B2%E7%9F%A5a%2Cb%5Cin%20%5Cmathbb%7BZ%7D%0A%5C%5C%E8%AF%81%E6%98%8E%3A%E8%8B%A5%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Bab%2B1%7D%E6%98%AF%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%8C%E5%88%99%E4%BA%A6%E4%B8%BA%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B0%0A%5Cend%7Barray%7D%7D%20

以下是证明：

%5Cbegin%7Barray%7D%7Bc%7D%0A%E8%AE%BE%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Bab%2B1%7D%3Dk%EF%BC%8C%E5%81%87%E8%AE%BEk%E4%B8%8D%E4%B8%BA%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B0%EF%BC%8C%E8%80%83%E8%99%91%E5%8F%8D%E8%AF%81%EF%BC%9A%5C%5C%0A%E5%81%87%E8%AE%BE(a%2Cb)%E6%98%AF%E6%96%B9%E7%A8%8B%E7%9A%84%E4%B8%80%E4%B8%AA%E6%9C%80%E5%B0%8F%E8%A7%A3(a%2Bb%E7%9A%84%E5%80%BC%E6%9C%80%E5%B0%8F%EF%BC%89%EF%BC%8C%E9%82%A3%E4%B9%88%5C%5C%0Aa%5E2%2Bb%5E2%3Dkab%2Bk%0A%5C%5C%E4%BA%8E%E6%98%AF%E5%8C%96%E7%AE%80%E4%B8%BA%E4%BB%A5a%E4%B8%BA%E4%B8%BB%E5%85%83%E7%9A%84%E4%BA%8C%E6%AC%A1%E5%87%BD%E6%95%B0%E5%BE%97%3A%0A%5C%5Ca%5E2-bka%2Bb%5E2-k%3D0%0A%5C%5C%E7%94%A8%E9%9F%A6%E8%BE%BE%E5%AE%9A%E7%90%86%E5%8F%AF%E5%BE%97%EF%BC%9A%0A%5C%5Ca%2Ba'%3Dbk%0A%5C%5Caa'%3Db%5E2-k%0A%5C%5C%E4%B8%8D%E5%A6%A8%E8%AE%BEa%3Eb%0A%5C%5C%E5%88%99a'%3Dbk-a%5Cin%5Cmathbb%20Z%0A%5C%5C%E5%8F%88a'b%5Cgeq0%20%E4%B8%94%20b%5Cgeq%200%0A%5C%5C%E6%89%80%E4%BB%A5a%5Cgeq%200%0A%5C%5Ca'%3D%5Cfrac%7Bb%5E2-k%7D%7Ba%7D%0A%5C%5C%E5%9B%A0%E4%B8%BAk%E4%B8%8D%E4%B8%BA%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B0%EF%BC%8C%E6%89%80%E4%BB%A5b%5E2-k%E2%89%A00%0A%5C%5C%E6%89%80%E4%BB%A5a'%E2%89%A00%0A%5C%5C%E5%8D%B3a'%E2%89%A50%0A%5C%5C%E5%8F%88a'%3D%5Cfrac%7Bb%5E2-k%7D%7Ba%7D%EF%BC%9Cb-%5Cfrac%7Bk%7D%7Bb%7D%3Cb%0A%5C%5C%E6%89%80%E4%BB%A5a'%3Cb%0A%5C%5C%E9%82%A3%E4%B9%88(a'%2Cb)%E6%98%AF%E6%AF%94(a%2Cb)%E6%9B%B4%E5%B0%8F%E7%9A%84%E8%A7%A3%EF%BC%8C%E4%B8%8E(a%2Cb)%E7%9A%84%E6%9C%80%E5%B0%8F%E6%80%A7%E7%9F%9B%E7%9B%BE%E3%80%82%0A%5C%5C%E6%95%85k%3D%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Bab%2B1%7D%20%E4%B8%BA%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B0%0A%5C%5C%E8%AF%81%E6%AF%95%E3%80%82%0A%5Cend%7Barray%7D

经典的无穷递降法，关键在于是否能够借助k不为完全平方数的性质，让希望构造出来的更小解不为0，而可以在不等于0的情况下，让方程的解无限变小，从而产生矛盾。

高中生自行整理，如有错误，欢迎指出。


标签：数学

1988.IMO.6数论--无穷递降法的评论 (共 条)

分享到微博请遵守国家法律