质数的一种判别法及应用

A judging method of prime number and its application

证明了下述质数判别法 :设不大于m的所有质数组成的集为Pm={ 2 ,3 ,5… pπ(m) } ,ri 为m2 关于模 pi(i =1,2…π(m) )的最小剩余。Bi={ pin -ri|n∈N∧ (pin -ri<2m + 1) } ,则m2 +x是质数的充分必要条件是x属于A与∪π(m )i =1Bi 的差集 ,其中A ={ 1,2 ,… 2m} ,0 <x≤ 2m。

This article gives a new judging method of prime number.If A=｛1,2,…,2m｝ and P n=｛p 1,p 2,…,p π(m)｝,then the full and essential condition of m 2+x in (m 2,(m+1) 2) is that prime numbers would be x∈A\ ∪ π(m)i=1B i, in which B i=｛p in-r i n∈N∧(p in-r i)≤2m｝ and r i≡m 2(mod p i).