A Search Procedure and Lower Bound for Odd Perfect Numbers
نویسندگان
چکیده
منابع مشابه
A Lower Bound for the Set of Odd Perfect Numbers
It is proved here that if n is odd and perfect, then n > 10s0. Whether or not the set of odd perfect numbers is empty is still an open, and apparently very difficult, question. However, many properties of the elements of this set have been determined. For example, it is well known that if n is both odd and perfect, then (1) n = po Pi • • • p, where the/», are distinct primes, pa = a0 = 1 (mod 4...
متن کاملAn Upper Bound for Odd Perfect Numbers
If N is an odd perfect number with k distinct prime factors then we show that N < 2 k . If some of the small prime factors of N are known then this bound can be further improved.
متن کاملImproved Techniques for Lower Bounds for Odd Perfect Numbers
If N is an odd perfect number, and q \\ N, q prime, k even, 2k then it is almost immediate that N > q .We prove here that, subject to certain conditions verifiable in polynomial time, in fact N > q ' . Using this and related results, we are able to extend the computations in an earlier paper to show that N > 10300 .
متن کاملOdd Perfect numbers
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] ...
متن کاملSieve methods for odd perfect numbers
Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 108 < p < 101000. These results are gene...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1973
ISSN: 0025-5718
DOI: 10.2307/2005529