Erds猜想对某些素数的结论及其它

Some Results on A Erdos Conjecture and Another Problem

本文证明了,对一些特殊类型的素数P,有一素数q<P,是P的原根。即,对这些素数P,Erdos猜想是肯定的。其次,我们完善了作者的结论,证明了:任意n个两两互素的,都不是完全平方数的n个自然数a_1,a_2,…,a_n,必存在无限多个奇数p,使a_1,a_2,…,a_n都是模p的二次非剩余。

In this paper, we have proved some results on the Erdos' conjecture, that is, if p is large enough,there is always a prime q<p so that q is a primitive root (mod p), and we have also discussed a problem about quadratic nonresidues mod p. Theorem 1 For the prime in one of the following forms 2~m+1,2r+1, 4r+1(>8103),8r+1(>162753),in which r is an old prime, the Erdos conjecture is affirmitive. Theorem 2 letc(p) denotes the set of priems that is quadratic nonresidues mod pin the numbers 1,2,…, p—1, when P>e^(50), we have |c(p)|>2/1 log p,where |c(p)| denotes the number of elements in c(p). Theorem 3 When an odd primep>e^(50), and satisfies(P-1)/2-(φ(p-1))/2≤1/2 log p, the Erdos, conjecture is affirmative for this P. Corollary For each natural number m, there exists a constant k(m), which depends only on m, so that, to the p-rime p=2~mr+1 in which r is an odd prime: the Erdos' conjecture is affirmitive. Theorem 4 Given n positive integers a_1, a_2,…,a_n, prime to each other and none of which is square, there exist in finite odd primes p(i≥1), a_1,a_2,…,a_n are all quadratic nonresidues mod p_i.