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| Perfect number | |
|---|---|
| Name | Perfect number |
| Field | Mathematics |
Perfect number A perfect number is an integer equal to the sum of its proper divisors (positive divisors excluding the number itself). This notion has been studied by Euclid, Nicomachus, Pythagoras, Leonhard Euler, and later mathematicians associated with Royal Society, Ecole Normale Supérieure, and institutions such as Princeton University and University of Göttingen. Perfect numbers connect to topics treated in works like Elements (Euclid), Arithmetica, Historia Mathematica, and modern journals such as Journal of Number Theory.
A perfect number is a positive integer n satisfying σ(n) = 2n, where σ denotes the sum-of-divisors function introduced in contexts by Leonhard Euler and formalized in treatises influenced by Euclid and Diophantus. Historically, the property was described by authors like Nicomachus in classifications found alongside writings attributed to Pythagoras and appeared in commentaries circulated through libraries such as the Library of Alexandria and later catalogues at Bodleian Library and Vatican Library. Formal modern treatments appear in texts from Cambridge University Press and lecture series at Massachusetts Institute of Technology.
Perfect numbers are rare and tightly constrained by results established by Euclid and Leonhard Euler. If n is perfect, then n is either even with a specific algebraic form or potentially odd; no odd examples are known. Even perfect numbers are related to Mersenne primes via the Euclid–Euler theorem; the theorem was developed within mathematical correspondence networks including figures at Paris, Berlin, and institutions like Royal Society. Properties of perfect numbers intersect with multiplicative functions studied by Carl Friedrich Gauss and explored in treatises by Sophie Germain and later analysts at ETH Zurich and University of Cambridge. Results involve congruences linked in papers in venues such as Annals of Mathematics and computational searches carried out by teams at Los Alamos National Laboratory and distributed projects connected to Great Internet Mersenne Prime Search.
Euclid proved that if 2^p − 1 is prime (a Mersenne prime), then 2^(p−1)(2^p − 1) is an even perfect number; this construction appears in Elements (Euclid) commentary and later proofs by Euler completed the converse. Known even perfect numbers correspond to known Mersenne primes discovered by a lineage of mathematicians and institutions including Marin Mersenne historically, and modern discoveries credited to researchers affiliated with University of Central Missouri and teams in projects like GIMPS. Even perfect numbers are triangular and hexagonal in figurate number theory traditions traced to Pythagoras; connections to polygonal numbers appear in expositions at Oxford University and Sorbonne. Each even perfect number ends with 6 or 28 in base 10 for the smallest cases demonstrated in writings preserved at British Library and exhibited in problem collections from École Polytechnique.
No odd perfect number has been found, and their nonexistence remains an open problem featured in problems lists compiled by David Hilbert and in presentations at conferences at Institute for Advanced Study. Partial results constrain their form: any odd perfect number would exceed enormous bounds, be divisible by primes with specific congruence conditions investigated by researchers at University of California, Berkeley and University of Illinois Urbana-Champaign, and would have prime power factorization properties studied in papers appearing in Proceedings of the American Mathematical Society. Work by mathematicians such as Pomerance, Nielsen, and Pace Nielsen establishes conditions (e.g., existence of a prime factor to the first power, restrictions on number of distinct prime factors) discussed in seminars held at Cornell University and Rutgers University.
The term and study originate in antiquity: Euclid presented results connecting perfect numbers with what later became called Mersenne primes, while Nicomachus classified numbers in writings circulated in the Library of Alexandria. Medieval commentators like Boethius transmitted these ideas into manuscripts owned by Vatican Library and later translated in Renaissance collections at Bibliothèque nationale de France. The modern name “perfect number” appears in translations and expositions produced by scholars in the milieu of Royal Society and in the work of number theorists at University of Göttingen; discussions continued through the eras of Carl Friedrich Gauss and Évariste Galois into contemporary research at Institute for Advanced Study and international conferences sponsored by International Mathematical Union.
Construction of even perfect numbers follows from identification of Mersenne primes; testing primality of 2^p − 1 uses algorithms such as the Lucas–Lehmer test developed by researchers affiliated with institutions like University of Cambridge and University of Washington. Computational searches leverage distributed computing projects like GIMPS and supercomputers at Lawrence Livermore National Laboratory and teams at University of Central Missouri. Theoretically, proofs by Euler and methods by Sophie Germain and Adrien-Marie Legendre underpin factorization-based constraints, while analytic techniques used by Paul Erdős and Atle Selberg yield density and divisor-sum estimates appearing in journals such as Acta Arithmetica.
Perfect numbers connect to concepts in recreational mathematics popularized by authors like Martin Gardner and in pedagogical material at Harvard University and Stanford University. They relate to Mersenne prime searches, cryptographic prime testing venues such as work at National Institute of Standards and Technology and theoretical studies in algebraic number theory pursued at Princeton University. Related concepts include abundant and deficient numbers treated by Nicomachus and revived by modern expositors at American Mathematical Society, amicable numbers studied by Pierre de Fermat and Leonhard Euler, and sociable chains examined in combinatorial number theory lectures at University of Toronto.