Odd perfect numbers have at least nine distinct prime factors
نویسنده
چکیده
An odd perfect number, N , is shown to have at least nine distinct prime factors. If 3 N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
منابع مشابه
Odd perfect numbers, Diophantine equations, and upper bounds
We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 1012P 2N < 24 k . Using this new bound, and extensive computations, we derive the inequality k ≥ 10. Introduction One of the oldest unsolved problems in mathematics is whether there exists an odd perfect number N ....
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It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10, (see Brent [1]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] ...
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ورودعنوان ژورنال:
- Math. Comput.
دوره 76 شماره
صفحات -
تاریخ انتشار 2007