Perfect Number | Definition & Facts - Encyclopedia Britannica

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External Websites

• Digital Commons at the Georgia Academy of Science - The Mathematical Magic of Perfect Numbers
• Dartmouth College - Department of Mathematics - On the radical of a perfect number
• Mathematics LibreTexts - Perfect Numbers and Mersenne Primes
• Boston University - Department of Mathematics and Statistics - Perfect Numbers
• International Journal of Research and Analytical Reviews - Perfect Numbers (PDF)
• Stanford University - Perfect numbers
• DigitalCommons@University of Nebraska - Lincoln - Perfect Numbers
• Wolfram MathWorld - Perfect Number
• CiteSeerX - Perfect Numbers and Finite Groups
• Cambridge University Press and Assessment - Note for the Perfect Numbers (PDF)
• University of the Pacific Libraries - Scholarly Commons - Perfect Numbers

perfect number mathematics Ask Anything Homework Help Written and fact-checked by Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... Britannica Editors Last updated Feb. 27, 2026 •History Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything Top Questions

What is a perfect number?

A perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6.

Who were the Pythagoreans and what was their interest in perfect numbers?

The Pythagoreans, founded around 525 bce, studied perfect numbers for their mystical properties.

What did Nicomachus of Gerasa contribute to the study of perfect numbers?

Nicomachus of Gerasa classified numbers as deficient, perfect, and superabundant based on the sum of their divisors, and attributed moral qualities to these definitions.

How did St. Augustine view the number six?

St. Augustine considered six a perfect number and believed God created all things in six days because the number is perfect.

What mathematical result concerning perfect numbers is found in Euclid’s Elements?

Euclid proved that if a sum of numbers in double proportion becomes a prime, the product of the sum and the last number is perfect.

perfect number, 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 (founded c. 525 bce) studied perfect numbers for their “mystical” properties.

The mystical tradition was continued by the Neo-Pythagorean philosopher Nicomachus of Gerasa (fl. c. 100 ce), 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, 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 bce), where he proves the proposition:

Britannica Quiz Numbers and Mathematics

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 showed that any even perfect number must be obtainable from Euclid’s formula. It is not known whether there are any odd perfect numbers.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Anoushka Pant.