奇完全数素因数的一个性质

A property of prime divisors of odd perfect numbers

利用高次Diophantine方程的结果讨论奇完全数素因数的性质。证明了:如果n是奇完全数,p是n素因数,r是p在n的标准分解式中的次数,则σ(n/pr)/pr≠qt其中σ(n/pr)是n/pr的约数和,q是奇素数,t是正奇数或者适合t≤6的正偶数。

Using some results on higher degree Diophantine equations, the properties of prime divisors of odd perfect numbers are discussed. If n is an odd perfect number, p is a prime divisor of n and r is the degree of p in the factorization of n, then the result σ（n/p^r）/p^r≠q^t is proved, where σ（n/p^r） is the sum of divisors of n/pr, q is an odd prime, t is either an odd positive integer or an even positive integer with t≤6.