Zynx Number(s) Theory(s)

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 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 that it deals with statements that are simple to understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's conjecture, which remains unsolved since the 18th century. German mathematician Carl Friedrich Gauss (1777–1855) once remarked, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."[1] It was regarded as the epitome of pure mathematics, with no applications outside mathematics, until the 1970s, when prime numbers became the basis for the creation of public-key cryptography algorithms, such as the RSA cryptosystem.

Definition:

Number Theory is the branch of mathematics that studies integers and their properties and relations.[2] The integers comprise a set that extends the set of natural numbers {1,2,3,…} to include number 0 and the negation of natural numbers {−1,−2,−3,…}. 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).[3][4]

Number theory is closely related to arithmetic and some authors use the terms as synonyms.[5] However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the real numbers.[6] In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships.[7] Traditionally, it is known as higher arithmetic.[8] By the early twentieth century, the term number theory had been widely adopted.[note 1] The term number means whole numbers, which refers to either the natural numbers or the integers.[9][10][11]

Elementary number theory studies aspects of integers that can be investigated using elementary methods such as elementary proofs.[12] Analytic number theory, by contrast, relies on complex numbers and techniques from analysis and calculus.[13] Algebraic number theory employs algebraic structures such as fields and rings to analyze the properties of and relations between numbers. Geometric number theory uses concepts from geometry to study numbers.[14] Further branches of number theory are probabilistic number theory,[15] combinatorial number theory,[16] computational number theory,[17] and applied number theory, which examines the application of number theory to science and technology.[18]

History:

In recorded history, knowledge of numbers existed in the ancient civilisations of Mesopotamia, Egypt, China, and India.[19] The earliest historical find of an arithmetical nature is the Plimpton 322, dated c. 1800 BC. It is a broken clay tablet that contains a list of Pythagorean triples, that is, integers (a,b,c). The triples are too numerous and too large to have been obtained by brute force.[20] The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity[21][1/2*(x−(1/x))]^2+1=[1/2(x+(1/x))]^2, which is implicit in routine Old Babylonian exercises.[22] It has been suggested instead that the table was a source of numerical examples for school problems.[23][note 2] Plimpton 322 tablet is the only surviving evidence of what today would be called number theory within Babylonian mathematics, though a kind of Babylonian algebra was much more developed.[24]

Although other civilizations probably influenced Greek mathematics at the beginning,[25] all evidence of such borrowings appear relatively late,[26][27] and it is likely that Greek arithmētikḗ (the theoretical or philosophical study of numbers) is an indigenous tradition.[28] The ancient Greeks developed a keen interest in divisibility. The Pythagoreans attributed mystical quality to perfect and amicable numbers. The Pythagorean tradition also spoke of so-called polygonal or figurate numbers.[29] Euclid devoted part of his Elements to topics that belong to elementary number theory, including prime numbers and divisibility.[30] He gave the Euclidean algorithm for computing the greatest common divisor of two numbers and a proof implying the infinitude of primes. The foremost authority in arithmētikḗ in Late Antiquity was Diophantus of Alexandria, who probably lived in the 3rd century AD. He wrote Arithmetica, a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f(x,y)=z^2 or f(x,y,z)=w^2. In modern parlance, Diophantine equations are polynomial equations to which rational or integer solutions are sought.

After the fall of Rome, development shifted to Asia, albeit intermittently. The Chinese remainder theorem appears as an exercise[31] in Sunzi Suanjing (between the third and fifth centuries).[32] The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections.[33][34] There is also some numerical mysticism in Chinese mathematics,[note 3] but, unlike that of the Pythagoreans, it seems to have led nowhere. While Greek astronomy probably influenced Indian learning[35] it seems to be the case that Indian mathematics is otherwise an autochthonous tradition.[36][37] Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences n≡a1modm1, n≡a2modm2 could be solved by a method he called kuṭṭaka, or pulveriser;[38] this is a procedure close to the Euclidean algorithm.[39] Āryabhaṭa seems to have had in mind applications to astronomical calculations.[35] Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the Pell equation. A general procedure for solving Pell's equation was probably found by Jayadeva; the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century).[40]

In the early ninth century, the caliph al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work.[41][42] Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī, 953 – c. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew[43] what would later be called Wilson's theorem. Other than a treatise on squares in arithmetic progression by Fibonacci no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica.[44]

French mathematician Pierre de Fermat (1607–1665) never published his writings but communicated through correspondence and wrote in marginal notes instead.[45] His contributions to number theory brought renewed interest in the field in Europe. He conjectured Fermat's little theorem, a basic result in modular arithmetic, and Fermat's Last Theorem, as well as proved Fermat's right triangle theorem.[2][46] He also studied prime numbers, the four-square theorem, and Pell's equations.[47][48]

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur[note 4] Christian Goldbach, pointed him towards some of Fermat's work on the subject.[49][50] This has been called the "rebirth" of modern number theory,[51] after Fermat's relative lack of success in getting his contemporaries' attention for the subject.[52] He proved Fermat's assertions, including Fermat's little theorem; made initial work towards a proof that every integer is the sum of four squares;[53] and specific cases of Fermat's Last Theorem.[54] He wrote on the link between continued fractions and Pell's equation.[55][56] He made the first steps towards analytic number theory.[57]

Three European contemporaries continued the work in elementary number theory. Joseph-Louis Lagrange (1736–1813) gave full proofs of the four-square theorem, Wilson's theorem, and developed the basic theory of Pell's equations. Adrien-Marie Legendre (1752–1833) stated the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation ax2+by2+cz2=0.[58] In his old age, he was the first to prove Fermat's Last Theorem for n=5.[59] Carl Friedrich Gauss (1777–1855) wrote Disquisitiones Arithmeticae (1801), which had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work the law of quadratic reciprocity[60] and developed the theory of quadratic forms. He also introduced some basic notation to congruences and devoted a section to computational matters, including primality tests.[61] He established a link between roots of unity and number theory.[62] In this way, Gauss arguably made forays towards Évariste Galois's work and the area algebraic number theory.

Starting early in the nineteenth century, the following developments gradually took place:

  • The rise to self-consciousness of number theory (or higher arithmetic) as a field of study.[63]

  • The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.

  • The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837),[64][65] whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.[66] The first use of analytic ideas in number theory actually goes back to Euler (1730s),[67][68] who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point;[69] Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).[70]

The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize.