Perfect Number

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perfect number

[′pər·fikt ′nəm·bər]
(mathematics)
An integer which equals the sum of all its factors other than itself.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Perfect Number

 

a positive integer that is equal to the sum of all its factors except itself. Examples of perfect numbers are 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.

As early as the third century B.C., Euclid showed that even perfect numbers could be obtained from the formula 2p–l(2p – 1) when p and 2p – 1 are prime numbers. Approximately 20 even perfect numbers have been found in this way. As of 1976, no odd perfect numbers are known, and their existence remains an open question. Perfect numbers were first investigated by the Pythagoreans, who ascribed a special mystical meaning to such numbers and combinations of such numbers.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

perfect number

equal in value to the sum of those natural numbers that are less than the given number but that also divide (with zero remainder) the given number. [Math.: EB, VII: 872]
Allusions—Cultural, Literary, Biblical, and Historical: A Thematic Dictionary. Copyright 2008 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Sandor, A note on S(n), where n is an even perfect number, Smarandache Notions J., 11(2000), No.
Further, for a subclass we will be able to determine F(n) for even perfect numbers n.
Every even perfect number n is a triangular number.
* Are there other even perfect numbers which are not of this form?
(2) Are there infinitely many even perfect numbers? The answer is probably yes.
There are no ordinary even perfect numbers which are S-perfect or completely S-perfect in sense 1.
Let [Z.sub.t][L.sub.n](m) be a loop ring, where t is an even perfect number of the form t = [2.sup.s] ([2.sup.s+1] - 1) for some s > 1, then ([alpha]) = [2.sup.s] + [2.sup.s] [g.sub.i] [member of] [Z.sub.t][L.sub.n](m) is an S-idempotent.
As t is an even perfect number, t must be of the form