关于孤立数的一些新结果

New Results of Anti-Sociable Numbers

完全数、相亲数以及孤立数一直是数论研究的一个重要课题.最近,在孤立数方面取得了一些进展,2000年,F.LUCA证明了Fermat数都是孤立数;2005年,乐茂华教授证明了2的方幂都是孤立数,用乐茂华教授的方法给出孤立数的一些新的结果:对于任意含有4w+1(w∈Z)型素因子的正整数n,设p为n的任意一个4w+1(w∈Z)型素因子,则在n2,p2n2,p4n2,p6n2里至少有一个是孤立数,因此可以证明孤立数在完全平方数里有正密度,另外也给出求解确定孤立数的方法.

Perfect number, amicable number and anti-sociable numbers are important topics in number theory. Recently, advances have been made in anti-sociable numbers. In 2000, F. LUCA proved that Fermat number are anti-sociable numbers, and in 2005, M.H. LE proved all powers of 2 are anti-sociable numbers. We have used the method of M.H. LE to obtain some new results of the anti-sociable numbers. For every integer n containing prime divisors that are 1 mod 4, let p mod 4 be an arbitrary prime divisor of n. There is at least one anti-sociable number in n^2, p^2n^2, p^4n^2, and p^6n^2. Therefore we can prove that anti-sociable numbers have positive density in perfect square numbers. We also give a method to find the exact anti-sociable numbers.