منابع مشابه
Odd 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 کاملOn Dickson's Theorem Concerning Odd Perfect Numbers
A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k 2 .
متن کاملOn the Nonexistence of Odd Perfect Numbers
In this article, we show how to prove that an odd perfect number with eight distinct prime factors is divisible by 5. A perfect number N is equal to twice the sum of its divisors: σ(N) = 2N . The theory of perfect numbers when N is even is well known: Euclid proved that if 2 − 1 is prime, then 2p−1(2p − 1) is perfect, and Euler proved that every one is of this type. These numbers have seen a gr...
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...
متن کاملOdd perfect numbers of a special form∗†
We shall show that there is a effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 2012
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972712000032