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对自然数N,若n~(1/2)是自然数,则称N是完全平方数。完全平方数有如下一条性质: 自然数N是完全平方数的充要条件是N的正约数的个数为奇数(注:这一性质的充分性部分曾作为八四年北京市的数学竞赛题)。证:充分性:设p是N的正约数,则p~(-1)N也是N的正约数,所以,N的正约数除n~(1/2)外,都是成对出
For the natural number N, if n ~ (1/2) is a natural number, then N is said to be a complete square number. The complete square number has one of the following properties: The necessary and sufficient condition for the natural number N to be a complete square number is that the number of positive divisors of N is an odd number (Note: The adequacy part of this property was used as a math contest in the 1984 Beijing Municipality. ). Proof: Sufficiency: Let p be a positive divisor of N, then p~(-1)N is also a positive divisor of N. Therefore, the positive divisor of N is divided into n pairs except for 1/2. Out