关键词: 孤立数, 同余, 相亲数

Abstract: 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 inn², p²n², p⁴n^2, and p⁶n². 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.

Key words: amicable number, congruence, anti-sociable number