循环小数的周期和数码和

如果素数p是102k-1u+1的一个因子,则说p在一k-类中,由此导出一个对素数的分类.设(b,10)=1且既约真分数a/b的循环节是q1q2…q2s,那么qi+qs+i=9当且仅当b的所有素因子都属于一k-类,这时a/b的数码和为9s.既约真分数a/3n+2的数码和为9(t-1)/2+r,这里t是a/3n+2的周期,r是a模9的最小非负剩余.如果1/p的周期等于p-1或(p-1)/2,那么p是一个素数.  

The period and numerals sum of repeating decimal

A prime p is said to be in a k-class if p is a prime divisor of 10 2k-1u+1,which leads to a classification for all primes.Let (b,10)=1 and the repetend of irreducible proper fraction a/b be q1q2…q2s,then qi+qs+i=9 if and only if all prime divisors of b belong to one k-class,the numerals sum of a/b is 9s in this case.The numerals sum of irreducible proper fraction a/3n+2 is 9(t-1)/2+r,where t is the period of a/3n+2 and r is the least non-negative residue of a modulo 9.If the period of 1/p equal to p-1 or (p-1)/2,then p is a prime.