Perfect number

A natural number n is (even perfect number ) called a perfect number if it is equal to the sum of σ * (n) is of all its (positive) divisor other than itself. An equivalent definition is: a perfect number n is a number that is half as large as the sum of all its positive divisors ( including herself ), ie, σ (n) = 2n. The smallest three perfect numbers are 6, 28 and 496 all known perfect numbers are derived straight and Mersenne primes. It is unknown whether there are odd perfect numbers. Even in ancient Greece were perfect numbers known, its main properties were treated in the elements of Euclid. Perfect numbers were often the subject zahlenmystischer and numerological meanings.

• 5.1 Abundant and deficient numbers
• 5.2 Friendly and sociable numbers
• 5.3 Pseudo full Arrived numbers
• 5.4 Numbers Weird or odd numbers
• 5.5 sublime figures
• 5.6 Super Perfect Numbers

• 6.1 Boëthius
• 6.2 Biblical exegesis
• 6.3 seal

Examples

In contrast to deficient and abundant numbers perfect numbers are very rare. The smallest known since ancient examples of perfect numbers are 6, 28, 496 and 8128 (follow- A000396 in OEIS ):

• The proper divisors of 6 are 1, 2 and 3 your total is.
• The proper divisors of 28 are 1, 2, 4, 7 and 14 is your total.
• The proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and 248 your total is.
• The proper divisors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032 and 4064th Their sum is.

Calculation of perfect numbers

Already Euclid noted that the first four perfect numbers from the formula

Can be calculated:

• For n = 2: = 6 = 1 2 3
• For n = 3: = 28 = 1 2 4 7 14
• For n = 5 = 496 = 1 2 4 8 16 31 62 124 248
• N = 7: = 8128 = 1 2 4 8 16 32 64 127 254 508 1016 2032 4064

The first 10 perfect numbers are:

Euclid proved that always is a perfect number, if a prime number is, these are the so-called Mersenne primes. Nearly 2,000 years later, Leonhard Euler proved that all even perfect numbers can be generated in this way; the Mersenne primes are 48 known so far, namely n in the following: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1,279, 2,203, 2,281, 3,217, 4,253, 4,423, 9,689, 9,941, 11,213, 19,937, 21 701, 23 209, 44 497, 86 243, 110 503, 132 049, 216 091, 756 839, 859 433, 1257787, 1398269, 2976221, 3021377, 6972. 593, 13,466,917, 20,996,011, 24,036,583, 25,964,951, 30,402,457, 32,582,657, 37,156,667, 42,643,801, 43,112,609, 57,885,161.

It is unknown whether there are infinitely many perfect numbers. In addition, it is unknown whether there are odd perfect numbers. It is known, however, that such a number, if it exists, is greater than 101500, and at least 8 (or 11, if the number is not divisible by 3 ) has different prime divisors.

Other properties of perfect numbers

The sum of the reciprocal divider

The sum of the reciprocals of all the divisors of a perfect number n (including the number itself ) returns 2

Example:

Representation of Eaton (1995, 1996)

Every even perfect number n> 6 has the representation

Conversely, you do not get to every natural number j is a perfect number.

Examples:

Odd perfect numbers

It is believed that there are no odd perfect numbers. If there is such a number yet, it has the following properties:

• It has the form or a natural number (that is, leaves a remainder of 1 when dividing by integer 12 or a group of 9 in integer division by 36).
• It has at least six prime factors
• It has at least 9 prime factors, if it is not divisible by 3
• It is less than 109118, then it is divisible by evenly divisible, which is a prime number larger than 10500, and has at least 8 different prime divisors (or at least 11, if not the number is divisible by 3 ).

Sum of the first odd natural numbers to the third power

With the exception of 6 can be any even perfect number n, with the appropriate natural number k as

Examples:

Sum of the first natural numbers

Every even perfect number n can be represented with a suitable natural number k as

Or in other words: Every even perfect number is a triangular number. As mentioned above, k is always a Mersenne prime.

Examples:

Generalization of perfect numbers

A k- perfect number is a number whose sum of its proper divisors yields k times the number himself. The perfect numbers are then exactly the - perfect numbers. All k- perfect numbers with k ≥ 2 are particularly abundant numbers.

Example:

Relationship with other classes of numbers

Abundant and deficient numbers

Abundant numbers are those natural numbers n, is in which the sum of proper divisors σ * (n ) is greater than the number itself. Deficient numbers are those integers for which this sum is smaller than the number itself.

The smallest abundant number is 12, the divisor is calculated as the sum.

Friendly and sociable numbers

Two different natural numbers, in which the sum of proper divisors σ * of the first number, the second and the first is the second number, called a friendly pair of numbers. The smaller of them is abundant and the greater is deficient.

Example:

If more than two natural numbers needed to come back again in this way the initial number, it is called sociable numbers (English sociable numbers ).

Example 5 sociable numbers:

Pseudo full Arrived numbers

A natural number is called pseudo fully come when they can be represented as the sum of several different real divider.

Example:

All pseudo- perfect numbers are either perfect or abundant.

A proper subset of the pseudo- perfect numbers are the primary pseudo- perfect numbers: Let be a composite number and the amount of prime divisors of. The number is called primary pseudo fully come when:.

Equivalent to the following characterization: A composite number with the set of prime divisors if and only pseudo fully come primarily when:. This reflects the close relationship of the primary pseudo- perfect numbers to the Giuga numbers that are characterized by.

The smallest known primarily pseudo- perfect numbers are:

• 2
• 6 = 2 x 3
• 42 = 2 × 7 × 3
• 1806 = 2 × 3 × 7 × 43
• 47 058 = 2 × 3 × 11 × 23 × 31
• 2214502422 = 2 × 3 × 11 × 23 × 31 × 47 059
• 52,495,396,602 = 2 × 3 × 11 × 17 × 101 × 149 × 3109
• 8.490.421.583.559.688.410.706.771.261.086 = 2 × 3 × 11 × 23 × 31 × 47 059 × 2217342227 × 1,729,101,023,519

Properties of the primary pseudo perfect numbers:

• All primary pseudo- perfect numbers are square-free.
• The number 6 is the only primary pseudo -perfect number, which is also perfect. All other primary pseudo- perfect numbers are abundant.
• There are only a finite number of primary pseudo perfect numbers with a given number of prime factors.
• It is not known whether there are infinitely many primary pseudo -perfect numbers.

Weird Numbers or odd numbers

A natural number n is called weird ( in German " strange " ) when abundant, but is not pseudo fully come. So you can not be represented as the sum of some of its proper divisors, although the total sum of their proper divisors exceed the number n.

Example: The number 70 is the smallest odd number. You can not be written as the sum of numbers from the divider set {1, 2, 5, 7, 10, 14, 35}. The next odd numbers are 836, 4030, 5830, 7192, 7912, 9272, 10430th

Features:

• There exist infinitely many odd numbers
• All known strange numbers are. It is unknown whether an odd odd number exists.

Exalted numbers

If both the divisor number and the sum of the divisors of a natural number perfect numbers, then is called the sublime. Currently (2010) only two raised figures are known: the 12 and a number with 76 digits. See the main article sublime number.

Super Perfect Numbers

When re- forms of the sum of a natural number, the divider dividing the sum and said second sum divider is twice as large as, that holds, then called a super perfect number.

• The number 2 has the divisor sum of 1 2 = 3, 3, the divisor sum 1 3 = 4 There is, 2 is super perfect.
• The perfect number 6 has the divisor sum 1 2 3 6 = 12, 12, the divisor sum 1 2 3 4 6 12 = 28 Therefore, 6 is not super perfect.

Perfect numbers in Late Antiquity and the Middle Ages

Boëthius

The arithmetic properties of perfect numbers and sometimes their interpretation arithmological belong in late antiquity to the arithmetic curriculum and passed by Boethius in his Institutio arithmetica that is largely based in turn on Nicomachus of Gerasa, in the Latin Middle Ages. Following his Greek original treated Boëthius the perfect numbers ( numerical perfecti secundum partium aggregationem ) as a subspecies of even numbers ( numerical pares ) and explains her on Euclid returning calculation principle in such a way that the links in the series of the even -even numbers ( numerical pariter pares: 2n) are added to each other until their sum is a prime number: multiply these prime with the most recently added row element, the result is a perfect number. Boëthius performs this calculation in the different steps for the first three perfect numbers 6, 28 and 496 and mentioned also the fourth perfect number 8128th On this finding is based on Boëthius the complementary observation to the legality of the perfect numbers, that they were to appear each decade ( power of ten ) exactly once and in this case ended up in the " Einern " refer respectively to 6 or 8. The statements of Boëthius formed in the following centuries, the sum of the arithmetic knowledge of the perfect numbers, which was passed in the treatises De arithmetica, in encyclopedias as the Etymologiae Isidore and other didactic works more or less complete, but without significant additions until with the discovery of the fifth perfect number (33550336) was recognized in the 15th century that the assumption on the uniform distribution on the ' decades ' is incorrect

While Boëthius remains largely limited in treating other payment methods on the arithmetic curriculum, him the perfect numbers give cause for further ethical considerations in which they the abundant (plus quam perfecti, also called superflui or abundantes ) and the deficit figures ( inperfecti, also deminuti or indigentes called ) are compared: while these latter two payment methods are similar to the human vices because they like these are very common right and submit to no particular order, the ' perfect numbers ' as the virtue behavior by the right measure, the mean between excess and deficiency, preserve, are extremely rare and subject themselves to a fixed order. Boëthius indicated at the same time an aesthetic preference of perfect numbers when he abundant with monsters from mythology such as the three-headed Geryon compares while the deficient with deformities compares who, like the one-eyed Cyclops, characterized by too little natural body parts are. In these comparisons, which already takes Boëthius from its Greek original, is in the background the idea that a number a of links ( partes), has composite body so that only the perfect numbers the links of the number in a balanced proportion to their bodies are.

Biblical exegesis

Your actual importance for the medieval tradition unfolded the perfect numbers in the Bible exegesis, where the interpretation of the six days of creation in which God completed the work of his creation ( " consummavit " in the Vetus Latina, " perfecit " in the Vulgate of Jerome ) the starting point made to establish a connection between the arithmetic ' perfection ' of the number six and the perfection of the divine work of creation. The number six was a prime example for the illustration of the view that the divine creation is ordered by measure, number and weight of this tradition. Decisive for the Latin world here was Augustine, who in turn evolved approaches of predecessors from the Alexandrian exegesis. Augustine has very often expressed in his exegetical and homiletic works for ' perfection ' of the number six, most extensively in his commentary De genesi ad litteram, where he not only explains the arithmetic facts and discusses the theological question of whether God the number six because of her perfection elected or her only gave this perfection by his choice, but in addition also demonstrated in the works of creation, that compliance with the six- speed reflected by their ' parts ' ( partes) 1, 2 and 3, and on the nature of the works of creation and a latent ' ordo 'corresponds to the creation

• The first day of creation with the creation of light, which also implies the creation of the heavenly intelligences, for Augustine, is as one day alone
• First and that on the second day of creation the ' upper range ', the firmament of heaven, and on the third day of creation of the ' lower area ', the dry land and: In him the two days where the world building, the fabrica mundi, was created following the sea.
• The last three days once again form a group by themselves, because to them those creatures were created to move in this ' fabrica mundi ' and populate it and adorn should be: on the fourth day first again at the top of the celestial bodies, sun, moon and stars, on the fifth day at the ' lower area ' the animals of the water and the air, and on the sixth day after all the animals of the country and as the most perfect work last man.

The perfection of the number six, which also appeals to Augustine as triangular number, results from these factual interpretation in two ways: on the one hand in the sequence of 1 2 3 days, on the other hand, but also by the fact that the work of the first day for no particular upper or lower area is assigned ( here symbolized by letter A ), the works of the following days, however, in each case either belong to the upper (B) or the bottom (C ) region, so that also in this respect again a perfect order of 1, 2 and 3 results in days with the distribution of A BC BCC.

Usually not perfectus with this detailed interpretation of the latent ' ordo ', but at least in the general interpretation as the arithmetic numerus was this understanding of the six -day work to the public domain of medieval exegesis and the starting point for the interpretation and almost all other occurrences of ways the six numbers in the Bible and the history of salvation. Among other things, in the interpretation deriving from the days of creation six world ages (Adam, Noah, Abraham, David, Babylonian captivity, Christ ) - which in turn than two " before the law " ( ante legem ) when three " under the law" ( sub lege) and as an age of grace (sub gratia ) were interpreted - in the interpretation of the six ages of man and in the interpretation of Holy Week - in which met on the sixth day from the sixth hour, the Passion of Christ - and many others biblical and extra-biblical Senare more.

Seal

This is followed also medieval poet latched seamlessly onto sometimes by laying the arithmetic understanding in his bibelexegetischen content imprinting for the development of their work is based. So Alcuin has a metrical poem in six stanzas of six verses to Gundrada, a relative of Charlemagne, written and illustrated in an accompanying prose statement that he chose the number six, so as to promote the moral ' perfectio ' of the recipient:

Alcuin students Rabanus Maurus has not only made in a similar way in several shorter poems such relationships to ' perfectio ' the number six, but also in his poetic masterpiece, the Liber de laudibus sanctae crucis, the overall structure of the ' perfectio ' of the aligned 28. This work consists of 28 character poems ( carmina figurata ), each of which a prose explanation and in the second book is accompanied by a paraphrase in prose. The figure poems themselves are written in hexameters of the poem within each having the same number of letters and written in the manuscripts without spaces between words, so that the metrical text is published as a rectangular block. Within this block, then individual letters are highlighted and orbits, which in turn again into new texts, so-called ' versus intexti ', can be put together. In prose statement on the 28th and last of these figures Hrabanus then points on the reasons for his choice of the number 28:

As in modern times Burkhard Taeger (1972 ) has discovered attacks the arithmetic understanding of number even deeper into the formal structure of the work. For one divides the 28 figure poems by number of letters per verse, then a grouping of 1, 2, 4, 7 and 14 results in poems, so that even in the inner structure of the work, the ' perfect ' compliance with the 28 by their partes reflects.

Evidence of poetic adaptations of the underlying numerical reasoning can also be in the later Middle Ages, see, and also in the visual arts, where one has to make do usually without explanatory additions to the construction of the works, one can assume that about 28 Giotto's frescoes on the life of St. Francis want to seal in the Upper Basilica of Assisi by their number the perfection of the saints and the Christ-likeness of his life.