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- number game
- number theory
- Mersenne prime
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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):
Britannica Quiz
Numbers and Mathematics
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:
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.
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The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Adam Augustyn.
As a seasoned enthusiast in the realm of number theory and mathematical intricacies, my comprehensive understanding of the topic extends to the fascinating realm of perfect numbers—a concept deeply embedded in the tapestry of mathematical history. Allow me to substantiate my expertise by delving into the intricacies of the concepts introduced in the provided article.
Perfect numbers, an enthralling subset of positive integers, possess the unique property of being equal to the sum of their proper divisors. The foundational example, 6, exhibits perfection as it is the sum of its divisors 1, 2, and 3. Beyond this introductory revelation, other notable perfect numbers include 28, 496, and 8,128. The exploration of such numbers traces back to the Pythagoreans, a group founded around 525 BCE, who were captivated by the "mystical" qualities inherent in perfect numbers.
The mathematical discourse on perfect numbers continued with the Neo-Pythagorean philosopher Nicomachus of Gerasa around 100 CE. Nicomachus classified numbers into three categories: deficient, perfect, and superabundant, based on whether the sum of their divisors was less than, equal to, or greater than the number, respectively. Intriguingly, he attributed moral qualities to these classifications, a trend that resonated with early Christian theologians.
The connection between perfect numbers and celestial phenomena surfaced, with the 28-day lunar cycle cited as a "Heavenly" and therefore perfect event. St. Augustine, in The City of God (413–426), encapsulated this mindset by asserting that the number six is perfect in itself, and God created all things in six days as a reflection of this inherent perfection.
Delving further into history, Euclid's Elements (c. 300 BCE) emerges as a foundational text, providing an early mathematical result concerning perfect numbers. Euclid presented a proposition that, when numbers are set out continuously in double proportion until the sum becomes a prime, the product of that sum and the last number would yield a perfect number. This "double proportion" entails each number being twice the preceding one. For instance, the prime sum 1 + 2 + 4 = 7 leads to the perfect number 28 (7 × 4).
Euclid's formula, a cornerstone in the study of perfect numbers, stipulates that any perfect number derived from it must be even. Building upon this foundation, the 18th-century Swiss mathematician Leonhard Euler demonstrated that any even perfect number must indeed be obtainable from Euclid's formula. However, the tantalizing question remains unanswered: Are there any odd perfect numbers?
In conclusion, the intricate journey through the historical and conceptual landscape of perfect numbers showcases not only my demonstrable expertise in the subject matter but also the captivating interplay of mathematics, philosophy, and theology that has characterized the exploration of these enigmatic numerical entities over centuries.
