# A 6 INTEGERS 12 A ( 2012 ) : John Selfridge Memorial Issue ON ODD PERFECT NUMBERS AND EVEN 3 - PERFECT NUMBERS
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چکیده
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
منابع مشابه
A 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...
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This seems simple enough, but let’s play with this definition. The Pythagoreans, an ancient sect of mathematical mystics, said that a number is perfect if it equals the sum of its positive integral divisors, excluding itself. For example, 6 = 1 + 2 + 3and 28 = 1 + 2 + 4 + 7 + 14are perfect numbers. On the other hand, 10 is not perfect because 1 + 2 + 5 = 8, and 12 is not perfect because 1+2+3+4...
متن کاملOn Perfect and Near-perfect Numbers
We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation 12 = 1 + 2 + 3 + 6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpiński. We discuss some rules for generating near-perfe...
متن کاملPerfect Numbers in ACL2
A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 6= 1 + 2 + 3 + 4 + 6 is not perfect. An ACL2 theory of perfect numbers is developed and used to prove, in ACL2(r), this bit of mathematical folklore: Even if there...
متن کاملPrime-perfect Numbers
We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1, x] satisfies estimates of the form exp((log x) log log log )...
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تاریخ انتشار 2012