In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ... In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. ... An Achilles number is a number that is powerful but not a perfect power. ... In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function &#963;(n)) is equal to 2n _ 1. ... In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function &#963;(n)) is equal to 2n + 1. ... In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. ... In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural number n for which the equality n = 1 + k(&#963;(n) &#8722; n &#8722; 1) holds, where &#963;(n) is the divisor function (i. ... A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. ... In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ... In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a natural number that has no semiperfect proper divisor. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a highly abundant number is a certain kind of natural number. ... In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. ... In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ... In mathematics, a superior highly composite number is a certain kind of natural number. ... In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ... The term weird number also refers to a phenomenon in twos complement arithmetic. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ... In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number. ... A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. ... A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents). ... An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents). ... An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... ... The integers are commonly denoted by the above symbol. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). Look up six in Wiktionary, the free dictionary. ... 28 (twenty-eight) is the natural number following 27 and preceding 29. ... Four hundred and ninety-six is the natural number following four hundred and ninety-five and preceding four hundred and ninety-seven. ... Cardinal eight thousand one hundred [and] twenty-eight Ordinal 8128th (eight thousand one hundred [and] twenty-eighth) Factorization Divisors 2, 4, 8, 11, 16, 22, 32, 44, 64, 127, 254, 508, 1,016, 2,032, 4,064 Roman numeral Binary 1111111000000 Octal 17700 Duodecimal 4854 Hexadecimal 1FC0 8,128 is... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

These first four perfect numbers were the only ones known to Hellenistic mathematicians. Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ...

Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹(2ⁿ − 1): For other uses, see Euclid (disambiguation). ...

for n = 2:   2¹(2² − 1) = 6

for n = 3:   2²(2³ − 1) = 28

for n = 5:   2⁴(2⁵ − 1) = 496

for n = 7:   2⁶(2⁷ − 1) = 8128

Noticing that 2ⁿ − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹(2ⁿ − 1) gives an even perfect number whenever 2ⁿ − 1 is prime (Euclid, Prop. IX.36). In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

Ancient mathematicians made many assumptions about perfect numbers based on the four they knew, but most of those assumptions would later prove to be incorrect. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2¹¹ − 1 = 2047 = 23 × 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:

• The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.
• The perfect numbers would alternately end in 6 or 8.

The fifth perfect number (33550336 = 2¹²(2¹³ − 1)) has 8 digits, thus refuting the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show the last digit of any even perfect number must be 6 or 8.

In order for 2ⁿ − 1 to be prime, it is necessary but not sufficient that n should be prime. Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. In mathematics, a Mersenne number is a number that is one less than a power of two. ... Marin Mersenne, Marin Mersennus or le Père Mersenne (September 8, 1588 – September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD realized that every even perfect number is of the form 2ⁿ⁻¹(2ⁿ − 1) where 2ⁿ − 1 is prime, but he was not able to prove this result.^[1] It was not until the 18th century that Leonhard Euler proved that the formula 2ⁿ⁻¹(2ⁿ − 1) will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the "Euclid-Euler Theorem". As of September 2007 only 44 Mersenne primes are known,^[2] which means there are 44 perfect numbers known, the largest being 2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits. (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] or Persian[2] Iraqi Muslim polymath[3][4] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

The first 39 even perfect numbers are 2ⁿ⁻¹(2ⁿ − 1) for

n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 (sequence A000043 in OEIS)

The other 5 known are for n = 20996011, 24036583, 25964951, 30402457, 32582657. As of 2006 it is not known whether there are others between them. The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... 2006 is a common year starting on Sunday of the Gregorian calendar. ...

It is still uncertain whether there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS distributed computing project. In set theory, an infinite set is a set that is not a finite set. ... The Great Internet Mersenne Prime Search, or GIMPS, is a collaborative project of volunteers, who use Prime 95 and MPrime, special open source software that can be downloaded from the Internet for free, in order to search for Mersenne prime numbers. ...

Since any even perfect number has the form 2ⁿ⁻¹(2ⁿ − 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2ⁿ − 1. Furthermore, any even perfect number except the first one is the sum of the first 2^(n−1)/2 odd cubes: A triangular number is the sum of the n natural numbers from 1 to n. ...

Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist.^[3] Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would imply that there are no odd perfect numbers. One of the top number theorists of our time, Carl Pomerance received his PhD from Harvard University in 1972 and immediately joined the faculty at the University of Georgia, becoming full professor in 1982. ... In computer science, besides the common use as rule of thumb (see heuristic), the term heuristic has two well-defined technical meanings. ... In mathematics, Ores harmonic numbers, defined by O. Ore in 1948, are defined as those positive integers for which the harmonic mean of its positive divisors is an integer. ...

Any odd perfect number N must satisfy the following conditions:

• N > 10³⁰⁰. A search is on to prove that N > 10⁵⁰⁰ is also required. ^[4]
• N is of the form

where:

• q, p₁, …, p_k are distinct primes (Euler).
• q ≡ α ≡ 1 (mod 4) (Euler).
• The smallest prime factor of N is less than (2k + 8) / 3 (Grün 1952).
• The relation e₁≡e₂...≡e_k ≡ 1 (mod 3) is not satisfied (McDaniel 1970).
• Either q^α > 10²⁰, or > 10²⁰ for some j (Cohen 1987).
• N < (Nielsen 2003).

• The largest prime factor of N is greater than 10⁸ (Takeshi Goto and Yasuo Ohno, 2006).
• The second largest prime factor is greater than 10⁴, and the third largest prime factor is greater than 100 (Iannucci 1999, 2000).
• N has at least 75 prime factors; and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors (Nielsen 2006; Kevin Hare 2005).

Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Minor results

Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist. There are a number 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: Richard K. Guy is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ... The law of small numbers may refer to the specific features of the Poisson distribution, as in the book The Law of Small Numbers by Ladislaus Bortkiewicz; or the tendency for an initial segment of data to show some bias that drops out later (one example in number theory being...

• An odd perfect number is not divisible by 105 (Kühnel 1949).
• Every odd perfect number is of the form 12m + 1 or 36m + 9 (Touchard 1953; Holdener 2002).
• The only even perfect number of the form x³ + 1 is 28 (Makowski 1962).
• A Fermat number cannot be a perfect number (Luca 2000).
• By dividing the definition through by the perfect number N, the reciprocals of the factors of a perfect number N must add up to 2:
  • For 6, we have 1 / 6 + 1 / 3 + 1 / 2 + 1 / 1 = 2;
  • For 28, we have 1 / 28 + 1 / 14 + 1 / 7 + 1 / 4 + 1 / 2 + 1 / 1 = 2, etc.
• The number of divisors of a perfect number (whether even or odd) must be even, since N cannot be a perfect square.
  • From these two results it follows that every perfect number is an Ore's harmonic number.

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... The reciprocal function: y = 1/x. ... In mathematics, Ores harmonic numbers, defined by O. Ore in 1948, are defined as those positive integers for which the harmonic mean of its positive divisors is an integer. ...

When e_i ≤ 2 for every i

• The smallest prime factor of N is greater than 739 (Cohen 1987).
• α ≡ 1 (mod 12) or α ≡ 9 (mod 12) (McDaniel 1970).

Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Related concepts

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and 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 amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number. In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... Numerology is any of many systems, traditions or beliefs in a mystical or esoteric relationship between numbers and physical objects or living things. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ...

By definition, a perfect number is a fixed point of the restricted sum-of-divisors function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. Look up definition in Wiktionary, the free dictionary. ... In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, an aliquot sequence is a recursive sequence which can be defined in the following way: if we write σ(n) = σ1(n) to be the divisor function normally, then, the aliquot sequence of k can be written: s0 = k sn = σ(sn−1) − sn−1 For example, the aliquot sequence... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

See also

• Perfection

For other uses, see Perfection (disambiguation). ...

Notes

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

References

The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

External links

• David Moews: Perfect, amicable and sociable numbers
• Perfect numbers - History and Theory
• Eric W. Weisstein, perfect number at MathWorld.
• List of Perfect Numbers at the On-Line Encyclopedia of Integer Sequences
• List of known Perfect Numbers All known perfect numbers are here.
• OddPerfect.org A projected distributed computing project to search for odd perfect numbers.