关于Legendre猜想的一个判定准则

A Criterion for the Legendre′s Conjecture

Legendre猜想是数论中一个与素数间隙有关的猜想,它断言对于每个正整数n,在2n和2(n+1)之间都有一个素数2.为此,文章首先研究了夹在2n和(n+1)之间的奇数,刻画了小于一个给定平方数的奇素数的特点.然后证明了如下的判定准则:若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)<n,则Legendre猜想成立;反之,若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)≥n,则Legendre猜想不成立.

Legendre＇s conjecture is a conjecture on prime gap in number theory. It states there is a prime between n^2 and （n ＋ 1 ）^2 for every positive integer n. In this paper a criterion for the conjecture was presented, to this end the odd primes less than a square was considered and a characterization of these odd primes was given and then a criterion for the Legendre＇s conjecture was also put forward： n∑r=3,r∈odd（[（n＋1）^2/2r]-[n^2-r^2/2r]）〈n,Legendre＇s conjecture holds, otherwise if n∑r=3,r∈odd（[（n＋1）^2/2r]-[n^2-r^2/2r]）≥n, then the Legendre＇s con-jecture fails.