平常数和例外数的2个性质

平常数的自然密度为1,例外数在自然数中的比例非常小,自然密度为0.利用平常数和例外数的定义及性质证明了在素因子给定的情况下,平常数只有有限个(因子重数很低的那些),例外数却有无限个(因子重数足够高之后).这表明一个数是否为例外数的决定因素是其因子重数的高低.

Two Properties of the Ordinary and Extraordinary Numbers

The natural density of the ordinary numbers is 1,and the proportion of the extraordinary numbers is very small,with the natural density 0.There are infinitely many integers with some given some primes.But there are only infinitely many ordinary numbers in them(those whose exponents of repetition factors are not very high),while the extraordinary numbers are infinitely many(when exponents of repetition factors are high enough).This indicates that it is the exponent of the repetition(prime) factors that determines whether the number is ordinary or not.