In number theorya perfect number is a positive integer that is equal to the sum of its proper positive divisorsthat is, the sum of its positive divisors excluding the number itself also known as its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself i.
Euclid also proved a
List the factors of 21 and 35 dating rule IX. Two millenia later, Euler proved that all even perfect numbers are of this form.
It is not List the factors of 21 and 35 dating whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: The next perfect number is Philo is followed by Origen and by Didymus the Blindwho adds the observation that there are only four perfect numbers that are less than 10, Commentary on Genesis
List the factors of 21 and 35 dating. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa.
This result is referred to as the Euclid—Euler theorem.
It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. Thus every even perfect number is a pernicious number. Note that every even perfect number is also a practical number c. It is unknown whether there is any odd perfect number,
List the factors of 21 and 35 dating various results have been obtained. InSylvester stated: All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare.
There are a of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy 's strong law of small numbers:. The sum of proper divisors gives various other kinds of numbers.
Numbers where the sum is less than the number itself are called deficientand where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology.
A pair of numbers which are the sum of each other's proper divisors are called amicableand larger cycles of numbers are sociable. A positive integer such that every smaller List the factors of 21 and 35 dating integer is a sum of distinct divisors of it is a practical number. A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors.
A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers. From Wikipedia, the free encyclopedia. For the film, see Perfect Number film.
History of the Theory of Numbers, Vol. Carnegie Institution of Washington. Between Arithmetic and Algebra Dordrecht: Kluwer Academic Publishers,pp. The History of Mathematics: Adventures in Mathematics, Mind, and Meaning. From Surreal Numbers to Magic Circles. Mathematical Association of America. Retrieved 9 May Retrieved 30 March Archived from the original on 21 February Retrieved 23 May International Journal of Number Theory.
Retrieved 13 August Note sur les nombres parfaits Texeira J. VIII11— An Introduction to Pure and Applied Mathematics. Joyce's website for a translation and discussion of this proposition and its proof.
Divisibility-based sets of integers. Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number. Untouchable Amicable Sociable Betrothed. Classes of natural numbers. Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star.
Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral. Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula. Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal. Centered pentachoric Squared triangular. Carmichael number Catalan pseudoprime Elliptic pseudoprime
List the factors of 21 and 35 dating pseudoprime Euler—Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Somer—Lucas pseudoprime Strong pseudoprime.
Amicable Betrothed Deficient Semiperfect. Aronson's sequence Ban Pancake number. Retrieved from " https: Divisor function Integer sequences Unsolved problems in mathematics.
All articles with dead external links Articles with dead external links from March Articles with permanently dead external links Articles with inconsistent citation formats Articles containing Ancient Greek-language text Articles containing potentially dated statements from January All articles containing potentially dated statements.
Are there any odd perfect numbers?