昨晚下課候，我閒來無事，看起一本書，裡頭提到了奇完美數．

我便開始研究了起來，結果發現一個有趣的定理．

奇完美數若存在，他的標準分解式，必然是這樣的形式

底數都是奇質數，指數恰好只有一個奇數次方，其餘都是偶數次方．

更深入研究後，發現一個引理，可以把這形式定的更精細點

奇完美數若存在，他的標準分解式，必然是質數╳完全平方數

本來我以為這是個獨特的發現，上網一查．

原來笛卡兒早在1638年就發現了，而且他跟我一樣．

一度認為這個發現，是有幫助的．我覺得好像有，但是要證明．

確實是要a great deal of time

在1638年，Descartes曾寫信給Mersenne，寫著─

Descartes, in a letter to Mersenne in 1638, wrote :-

... I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers...

    我想我可以由歐幾里得的公式證明，沒有不是偶數的完美數；但奇數的完美數必是，某一質數乘以一完全平方數，且其平方根為質數的合成。如22021是質數，9018009的平方根是由質數3，7，11，13的合成，所以22021*9018009=198585576189是完美數。…………………

問題：笛卡兒舉的例子乍看之下是合理的，其實是有問題的．

　　　那這個例子的錯誤在哪邊呢？