On Necessary Conditions for the Existence of Odd Perfect Numbers
نویسندگان
چکیده
منابع مشابه
Necessary Conditions For the Non-existence of Odd Perfect Numbers
We start with a result showing most odd cubes cannot be perfect numbers (see Theorem 1). Then we give a new proof of a special case of a result of Iannucci (see [IAN]) that shows that none of the even exponents in N ’s prime factorization can be congruent to 4 (mod 5) if 3|N (Theorem 2). We then extend that result by proving that certain sets of small primes, when taken to a large power, cannot...
متن کاملA Study on the Necessary Conditions for Odd Perfect Numbers
A collection of all of the known necessary conditions for an odd perfect number to exist, along with brief descriptions as to how these were discovered. This was done in order to facilitate those who would like to further pursue the necessary conditions for odd perfect numbers, or those who are searching for odd perfect numbers themselves. All past research into odd perfect numbers has been col...
متن کاملA Rationality Condition for the Existence of Odd Perfect Numbers
A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ(N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the charac...
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 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] ...
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ژورنال
عنوان ژورنال: Rocky Mountain Journal of Mathematics
سال: 1999
ISSN: 0035-7596
DOI: 10.1216/rmjm/1181071684