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Let (n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if (n )=2 n. It is well known that a number is even and perfect if and only if it has the form 2p 1(2p 1) where 2p 1i s prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed 106, and Hagis showed that the second largest must exceed 103. In this paper, we improve the latter bound. In particular, we prove the statement in the title of this paper.

The second largest prime divisor of an odd perfect number exceeds ten thousand