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Aliquot Sum
In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. Examples For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. (). The values of for are: :0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... Characterization of classes of numbers The aliquot sum function can be used to characterize several notable classes of numbers: *1 is the only number whose aliquot sum is 0. *A number is prime if and only if its aliquot sum is 1. *The aliquot sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abu Mansur Al-Baghdadi
Abū Manṣūr ʿAbd al-Qāhir ibn Ṭāhir bin Muḥammad bin ʿAbd Allāh al-Tamīmī al-Shāfiʿī al-Baghdādī (), more commonly known as Abd al-Qāhir al-Baghdādī () or simply Abū Manṣūr al-Baghdādī () was an Arab Sunni scholar from Baghdad. He was considered a leading Ash'arite theologian and Shafi'i jurist. He was an accomplished legal theoretician, man of letters, poet, prosodist, grammarian, heresiologist and mathematician. Life 'Abd al-Qahir al-Baghdadi was born and raised in Baghdad. He was a member of the Arab tribe of Banu Tamim. Ibn 'Asakir writes that Abu Mansur met the students of the companions of Imam al-Ashari and acquired knowledge from them. Among the Ashari imams of the third generation, he is the senior of al-Bayhaqi and the identical contemporary of Abu Dharr al-Harawi and Abu Muhammad al-Juwayni. Abu Mansur was brought in his youth by his father from Baghdad to Nishapur. In his new city is where he found his education. Abu Mansur was k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the Iterated function, iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer point, integer, rational point, rational, p-adic number, -adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. ''Global arithmetic dynamics'' is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while ''local arithmetic dynamics'', also called p-adic dynamics, p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers by a -adic field such as or and studies chaotic behavior and the Fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elemente Der Mathematik
''Elemente der Mathematik'' is a peer-reviewed scientific journal covering mathematics. It is published by the European Mathematical Society Publishing House on behalf of the Swiss Mathematical Society. It was established in 1946 by Louis Locher-Ernst, and transferred to the Swiss Mathematical Society in 1976. Rather than publishing research papers, it focuses on survey papers aimed at a broad audience. History The journal ''Elemente der Mathematik'' was founded in 1946 by Louis Locher-Ernst under the patronage of the Swiss Mathematical Society (SMG) to disseminate pedagogical and expository articles in mathematics and physics. Locher-Ernst outlined the scope and objectives—emphasising support for secondary and tertiary instruction—in a letter to the SMG president in August 1945 and at the autumn members' meeting in Fribourg later that year. Early editorial responsibilities were assumed by Locher-Ernst alongside Erwin Voellmy, Ernst Trost and Paul Buchner, while an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Of Auberive
William of Auberive () was a Cistercian monk and numerologist who served as the abbot of from 1165 until 1186.Uta-Renate Blumenthal, "Cardinal Albinus of Albano and the ''Digesta pauperis scolaris Albini'': Ms. Ottob. lat. 3057", ''Archivum Historiae Pontificiae'' 20 (1982): 7–49 (at p. 37, n. 124). Two of William's treatises—''De sacramentis numerorum a ternario usque ad duodenarium'' and ''Regule arithmetice''—and three of his letters have been published. Robert Earl Kaske, Arthur Groos and Michael W. Twomey, ''Medieval Christian Literary Imagery: A Guide to Interpretation'' (University of Toronto Press, 1988), p. 171. He belonged to a group of Cistercians whose work was an expansion on the work of Hugh of Saint-Victor allegorizing numbers in the Bible. His ''De sacramentis'' on the numbers three (3) through twelve (12) is a direct continuation of the ''Analytica numerorum'' of Odo of Morimond on the first three natural numbers. It was itself followed by the work of G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function ''σ''''z''(''n''), for a real or complex number ''z'', is defined as the sum of the ''z''th powers of the positive divisors of ''n''. It can be expressed in sigma notation as :\sigma_z(n)=\sum_ d^z\,\! , where is shorthand fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amicable Numbers
In mathematics, the amicable numbers are two different natural numbers related in such a way that the addition, sum of the proper divisors of each is equal to the other number. That is, ''s''(''a'')=''b'' and ''s''(''b'')=''a'', where ''s''(''n'')=σ(''n'')-''n'' is equal to the sum of positive divisors of ''n'' except ''n'' itself (see also divisor function). The smallest pair of amicable numbers is (220 (number), 220, 284 (number), 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992) . It is unknown if there are infinitely many pairs of amicable numbers. A pair of amicable numbers constitutes an aliquot sequence of Periodic sequence, pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Sequence
In mathematics, a periodic sequence (sometimes called a cycle or orbit) is a sequence for which the same terms are repeated over and over: :''a''1, ''a''2, ..., ''a''''p'', ''a''1, ''a''2, ..., ''a''''p'', ''a''1, ''a''2, ..., ''a''''p'', ... The number ''p'' of repeated terms is called the period ( period). Definition A (purely) periodic sequence (with period ''p''), or a ''p-''periodic sequence, is a sequence ''a''1, ''a''2, ''a''3, ... satisfying :''a''''n''+''p'' = ''a''''n'' for all values of ''n''. If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function. The smallest ''p'' for which a periodic sequence is ''p''-periodic is called its least period or exact period. Examples Every constant function is 1-periodic. The sequence 1,2,1,2,1,2\dots is periodic with least period 2. The sequence of digits in the decimal expansion of 1/7 is periodic with pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sociable Number
In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point. The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated. For example, on the image on the right: : Iterated functions are studied in computer science, fractals, dynamical systems, mathematics and renormalization group physics. Definition The formal definition of an iterated function on a set ''X'' follows. Let be a set and be a function. Defining as the ''n''-th iterate of , where ''n'' is a non-negative integer, by: f^0 ~ \stackrel ~ \operatorname_X and f^ ~ \stackrel ~ f \circ f^, where is the identity function on and denotes function composition. This notation has been traced to and John Frederick William Herschel in 1813. Herschel credited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime Number
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes, since they include two primes, or second numbers, by analogy with how "prime" means "first". Alternatively non-prime semiprimes are called almost-prime numbers, specifically the "2-almost-prime" biprime and "3-almost-prime" triprime Examples and variations The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case k=2 of the k- almost primes, numbers with exactly k prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach's Conjecture
Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than but remains unproven despite considerable effort. History Origins On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of Prime number#Primality of one, considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (""), in which Goldbach had remarked that the first of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
