# perfect number

perfect number
a positive number that is equal to the sum of all positive integers that are submultiples of it, as 6, which is equal to the sum of 1, 2, and 3. Cf. abundant number, deficient number.
[1350-1400; ME]

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a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128. The discovery of such numbers is lost in prehistory. It is known, however, that the Pythagoreans (Pythagoreanism) (founded c. 525 BC) studied perfect numbers for their “mystical” properties.

The mystical tradition was continued by the Neo-Pythagorean philosopher Nicomachus of Gerasa (fl. c. AD 100), who classified numbers as deficient, perfect, and superabundant according to whether the sum of their divisors was less than, equal to, or greater than the number, respectively. Nicomachus gave moral qualities to his definitions, and such ideas found credence among early Christian theologians. Often the 28-day cycle of the Moon around the Earth was given as an example of a “Heavenly,” hence perfect, event that naturally was a perfect number. The most famous example of such thinking is given by St. Augustine (Augustine, Saint), who wrote in The City of God (413–426):

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.

The earliest extant mathematical result concerning perfect numbers occurs in Euclid's Elements (c. 300 BC), where he proves the proposition:

If as many numbers as we please beginning from a unit [1] be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

Here “double proportion” means that each number is twice the preceding number, as in 1, 2, 4, 8, …. For example, 1 + 2 + 4 = 7 is prime; therefore, 7 × 4 = 28 (“the sum multiplied into the last”) is a perfect number. Euclid's formula forces any perfect number obtained from it to be even, and in the 18th century the Swiss mathematician Leonhard Euler (Euler, Leonhard) showed that any even perfect number must be obtainable from Euclid's formula. It is not known whether there are any odd perfect numbers.

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Universalium. 2010.

### Look at other dictionaries:

• Perfect number — Perfect Per fect, a. [OE. parfit, OF. parfit, parfet, parfait, F. parfait, L. perfectus, p. p. of perficere to carry to the end, to perform, finish, perfect; per (see {Per }) + facere to make, do. See {Fact}.] 1. Brought to consummation or… …   The Collaborative International Dictionary of English

• perfect number — n. a positive integer which is equal to the sum of all its factors, excluding itself: the first four perfect numbers are 6, 28, 496, and 8,128 …   English World dictionary

• Perfect number — In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect… …   Wikipedia

• perfect number — noun : an integer (as 6) the sum of whose divisors including 1 but excluding itself is equal to itself 28, which . 1 + 2 + 4 + 7 + 14, is a perfect number compare imperfect number * * * Math. a positive number that is equal to the sum of all… …   Useful english dictionary

• perfect number — noun A number that is the sum of all of its divisors except itself. The factors of 6 are 1, 2, 3 and 6, and 1 + 2 + 3 = 6, so 6 is a perfect number …   Wiktionary

• perfect number — per′fect num′ber n. math. a positive number that is equal to the sum of all positive integers that are submultiples of it, as 6, which is equal to the sum of 1, 2, and 3 • Etymology: 1350–1400 …   From formal English to slang

• perfect number — /pɜfəkt ˈnʌmbə/ (say perfuhkt numbuh) noun a number which is equal to the sum of its aliquot parts …   Australian-English dictionary

• perfect number — noun Date: 14th century an integer (as 6 or 28) the sum of whose integral factors including 1 but excluding itself is equal to itself …   New Collegiate Dictionary

• Multiply perfect number — In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k perfect (or k fold perfect) if and only if the sum of… …   Wikipedia

• Unitary perfect number — A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n / d share no common factors.) Some perfect numbers are… …   Wikipedia