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排列数P_n~r∈N。若r>1,则(P_n~r)~(1/2)是无理数。论证它,须从下面的有关引理开始。引理1 正整数与无理数之积是无理数。证令a是一个正整数,a是一个无理数。假设aα是有理数,就能表示成aα=q/p(p、q互质)(?)α=q/ap的左边是无理数等于右边是有理数的矛盾。所以,aα是无理数。引理2 若n为质数,则n~(1/2)是无理数。证假设n~(1/2)是有理数,就能表示成n~(1/2)=q/p(p、q互质)(?)np~2=q~2。即q~2能被n整除,而n是质数,因而q能被n整除。
The number of arrays P_n~r∈N. If r>1, (P_n~r)~(1/2) is an irrational number. To prove it, one must start with the following lemma. Lemma 1 The product of positive integers and irrational numbers is irrational. The warrant a is a positive integer and a is an irrational number. Assuming that aα is a rational number, it can be expressed as aα=q/p(p,q coprime)(?)α=q/ap. On the left is the contradiction that the irrational number equals the rational number on the right. Therefore, aα is an irrational number. Lemma 2 If n is a prime number, n~(1/2) is an irrational number. Assuming that n~(1/2) is a rational number, it can be expressed as n~(1/2)=q/p(p,q coprime)(?)np~2=q~2. That is, q~2 can be divisible by n, and n is a prime number, so q can be divisible by n.