Odd Perfect numbers
نویسنده
چکیده
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] who showed that if N is an odd perfect number with at most k prime factors, then N ≤ (4k)(4k)2 k2 . (1)
منابع مشابه
# A 6 INTEGERS 12 A ( 2012 ) : John Selfridge Memorial Issue ON ODD PERFECT NUMBERS AND EVEN 3 - PERFECT NUMBERS
An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there are no even 3-perfect numbers of the form 2aM , where M is odd and squarefree and a ≤ 718, besides the six known examples. –In memory of John Selfridge
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 کاملOdd perfect numbers have at least nine distinct prime factors
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.
متن کاملThe Imperfect Fibonacci and Lucas Numbers
A perfect number is any positive integer that is equal to the sum of its proper divisors. Several years ago, F. Luca showed that the Fibonacci and Lucas numbers contain no perfect numbers. In this paper, we alter the argument given by Luca for the nonexistence of both odd perfect Fibonacci and Lucas numbers, by making use of an 1888 result of C. Servais. We also provide a brief historical accou...
متن کاملNecessary 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 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.
متن کامل