奇完全数的两个猜想

Two Conjectures on Odd Perfect Numbers

运用初等方法讨论有关奇完全数的两个猜想.证明了:(i)如果n=p~αq_1^(2β_1)q_2^(2β_2)…q_s^(2β_s)是奇完全数,其中P,q_1,q_2,…,q_s是不同的奇素数,α,β_1,β_2,…,β_s是正整数,p≡α≡1(mood4),而且q_i≡-1(mod m)(i=1,2,…,s),m是大于2的正整数,则.1/2σ(p~α)必为合数;(ii)如果n=a^2~x+b^2~x,其中a,b,x是适合a>b,gcd(a,6)=1,2|ab的正整数,则当x≥log_2log_2log_2 a时,n不是奇完全数.

Using some elementary methods,two conjectures on odd perfect numbers are discussed.We prove that(i) Let m be a positive integer with m > 2.If n=p~αq_1^(2β_1)q_2^(2β_2)…q_s^(2β_s)is an odd perfect number,where p,q_1,q_2,…,q_s are distinct odd primes satisfying p ≡ 1(mod4) and q_i ≡-1(mod m) for i = 1,2,…,s,then 1/2σ(p~α) must be a compositive number,(ii) Let a,b,x be positive integers such that a > b,gcd(a,b) = 1 and 2 | ab.If x≥ log_2 log_2 log_2 α,then a^(2x) + b^(2x) is not an odd perfect number.