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 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) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." It was regarded as the example of pure mathematics with no applications outside mathematics until the 1970s, when it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms.

History

Number theory is the branch of mathematics that studies integers and their properties and relations. The integers comprise a set that extends the set of natural numbers $\$ to include number $0$ and the negation of natural numbers $\$. 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).

Number theory is closely related to arithmetic and some authors use the terms as synonyms. However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the real numbers. In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships. Traditionally, it is known as higher arithmetic. By the early twentieth century, the term ''number theory'' had been widely adopted.The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example, . In 1952, Davenport still had to specify that he meant ''The Higher Arithmetic''. Hardy and Wright wrote in the introduction to '' An Introduction to the Theory of Numbers'' (1938): "We proposed at one time to change he titleto ''An introduction to arithmetic'', a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." The term number means whole numbers, which refers to either the natural numbers or the integers.

Elementary number theory studies aspects of integers that can be investigated using elementary methods such as elementary proofs. Analytic number theory, by contrast, relies on complex numbers and techniques from analysis and calculus. 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. Further branches of number theory are probabilistic number theory, combinatorial number theory, computational number theory, and applied number theory, which examines the application of number theory to science and technology.

Origins

Ancient Mesopotamia

The earliest historical find of an arithmetical nature is a fragment of a table: Plimpton 322 (Larsa, Mesopotamia, c. 1800 BC), a broken clay tablet, contains a list of " Pythagorean triples", that is, integers $(a,b,c)$ such that $a^2+b^2=c^2$. The triples are too numerous and too large to have been obtained by brute force. The heading over the first column reads: "The of the diagonal which has been subtracted such that the width..."

The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity

$\left(\frac \left(x - \frac\right)\right)^2 + 1 = \left(\frac \left(x + \frac \right)\right)^2,$

which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by $c/a$, presumably for actual use as a "table", for example, with a view to applications.

It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own many centuries later. It has been suggested instead that the table was a source of numerical examples for school problems.. This is controversial. See Plimpton 322. Robson's article is written polemically with a view to "perhaps ..knocking limpton 322off its pedestal" ; at the same time, it settles to the conclusion that

..the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems .

Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics". 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.

Ancient Greece

Although other civilizations probably influenced Greek mathematics at the beginning, all evidence of such borrowings appear relatively late, Iamblichus, ''Life of Pythagoras'',(trans., for example, ) cited in . See also Porphyry, ''Life of Pythagoras'', paragraph 6, in Herodotus (II. 81) and Isocrates (''Busiris'' 28), cited in: . On Thales, see Eudemus ap. Proclus, 65.7, (for example, ) cited in: . Proclus was using a work by Eudemus of Rhodes (now lost), the ''Catalogue of Geometers''. See also introduction, on Proclus's reliability. and it is likely that Greek (the theoretical or philosophical study of numbers) is an indigenous tradition. Aside from a few fragments, most of what is known about Greek mathematics in the 6th to 4th centuries BC (the Archaic and Classical periods) comes through either the reports of contemporary non-mathematicians or references from mathematical works in the early Hellenistic period. In the case of number theory, this means largely Plato, Aristotle, and Euclid.

Plato had a keen interest in mathematics, and distinguished clearly between and calculation (). Plato reports in his dialogue '' Theaetetus'' that Theodorus had proven that $\sqrt, \sqrt, \dots, \sqrt$ are irrational. Theaetetus, a disciple of Theodorus's, worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. Aristotle further claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: ("They say Plato learned all things Pythagorean").

Euclid devoted part of his '' Elements'' (Books VII–IX) to topics that belong to elementary number theory, including prime numbers and divisibility. He gave an algorithm, the Euclidean algorithm, for computing the greatest common divisor of two numbers (Prop. VII.2) and a proof implying the infinitude of primes (Prop. IX.20). There is also older material likely based on Pythagorean teachings (Prop. IX.21–34), such as "odd times even is even" and "if an odd number measures dividesan even number, then it also measures divideshalf of it"., cited in: . This is all that is needed to prove that $\sqrt$ is irrational. Pythagoreans apparently gave great importance to the odd and the even. The discovery that $\sqrt$ is irrational is credited to the early Pythagoreans, sometimes assigned to Hippasus, who was expelled or split from the Pythagorean community as a result.Plato, ''Theaetetus'', p. 147 B, (for example, ), cited
in : "Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit;..." See also Spiral of Theodorus. This forced a distinction between ''numbers'' (integers and the rationals—the subjects of arithmetic) and ''lengths'' and ''proportions'' (which may be identified with real numbers, whether rational or not).

The Pythagorean tradition also spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th centuries).

An epigram published by Lessing in 1773 appears to be a letter sent by Archimedes to Eratosthenes. The epigram proposed what has become known as Archimedes's cattle problem; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation). As far as it is known, such equations were first successfully treated by Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

= Late Antiquity

=

Aside from the elementary work of Neopythagoreans such as Nicomachus and Theon of Smyrna, the foremost authority in in Late Antiquity was Diophantus of Alexandria, who probably lived in the 3rd century AD, approximately five hundred years after Euclid. Little is known about his life, but he wrote two works that are extant: ''On Polygonal Numbers'', a short treatise written in the Euclidean manner on the subject, and the '' Arithmetica'', a work on pre-modern algebra (namely, the use of algebra to solve numerical problems). Six out of the thirteen books of Diophantus's ''Arithmetica'' survive in the original Greek and four more survive in an Arabic translation. The ' is 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.

Asia

The Chinese remainder theorem appears as an exercise in '' Sunzi Suanjing'' (between the third and fifth centuries).The date of the text has been narrowed down to 220–420 AD (Yan Dunjie) or 280–473 AD (Wang Ling) through internal evidence (= taxation systems assumed in the text). See . (There is one important step glossed over in Sunzi's solution:''Sunzi Suanjing'', Ch. 3, Problem 26,
in :

6Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. ''Answer'': 23.

''Method'': If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When numberexceeds 106, the result is obtained by subtracting 105.

it is the problem that was later solved by Āryabhaṭa's Kuṭṭaka – see below.) The result was later generalized with a complete solution called ''Da-yan-shu'' () in Qin Jiushao's 1247 '' Mathematical Treatise in Nine Sections'' which was translated into English in early nineteenth century by British missionary Alexander Wylie. There is also some numerical mysticism in Chinese mathematics,See, for example, ''Sunzi Suanjing'', Ch. 3, Problem 36, in :

6Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. ''Answer'': Male.

''Method'': Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars f the Dipper 8 the eight winds, and 9 the nine divisions f China under Yu the Great If the remainder is odd, he sexis male and if the remainder is even, he sexis female.

This is the last problem in Sunzi's otherwise matter-of-fact treatise. but, unlike that of the Pythagoreans, it seems to have led nowhere.

While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an autochthonous tradition;Any early contact between Babylonian and Indian mathematics remains conjectural . in particular, there is no evidence that Euclid's ''Elements'' reached India before the eighteenth century. Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences $n\equiv a_1 \bmod m_1$, $n\equiv a_2 \bmod m_2$ could be solved by a method he called ''kuṭṭaka'', or ''pulveriser''; this is a procedure close to (a generalization of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations.

Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century).

Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.

Arithmetic in the Islamic golden age

In the early ninth century, the caliph al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the ''Sindhind'', which may or may not, and , cited in . be Brahmagupta's Brāhmasphuṭasiddhānta).
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 what would later be called Wilson's theorem.

Western Europe in the Middle Ages

Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—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''.

Early modern number theory

Fermat

Pierre de Fermat (1607–1665) never published his writings but communicated through correspondence instead. Accordingly, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes. Although he drew inspiration from classical sources, in his notes and letters Fermat scarcely wrote any proofs—he had no models in the area.

Over his lifetime, Fermat made the following contributions to the field:
* One of Fermat's first interests was perfect numbers (which appear in Euclid, ''Elements'' IX) and amicable numbers;Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in earlier times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean Nicomachus (c. 100 AD), who wrote a very elementary but influential book entitled ''Introduction to Arithmetic''. See . these topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.
* In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.
* Fermat's little theorem (1640): if ''a'' is not divisible by a prime ''p'', then $a^ \equiv 1 \bmod p.$Here, as usual, given two integers ''a'' and ''b'' and a non-zero integer ''m'', we write $a \equiv b \bmod m$ (read "''a'' is congruent to ''b'' modulo ''m''") to mean that ''m'' divides ''a'' − ''b'', or, what is the same, ''a'' and ''b'' leave the same residue when divided by ''m''. This notation is actually much later than Fermat's; it first appears in section 1 of Gauss's '. Fermat's little theorem is a consequence of the fact that the order of an element of a group divides the order of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo ''p'', that is, given ''a'' not divisible by a prime ''p'', there is an integer ''x'' such that $x a \equiv 1 \bmod p$); this fact (which, in modern language, makes the residues mod ''p'' into a group, and which was already known to Āryabhaṭa; see above) was familiar to Fermat thanks to its rediscovery by Bachet . Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.
* If ''a'' and ''b'' are coprime, then $a^2 + b^2$ is not divisible by any prime congruent to −1 modulo 4; and every prime congruent to 1 modulo 4 can be written in the form $a^2 + b^2$. These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent.
* In 1657, Fermat posed the problem of solving $x^2 - N y^2 = 1$ as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker. Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.
* Fermat stated and proved (by infinite descent) in the appendix to ''Observations on Diophantus'' (Obs. XLV) that $x^ + y^ = z^$ has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that $x^3 + y^3 = z^3$ has no non-trivial solutions, and that this could also be proven by infinite descent. The first known proof is due to Euler (1753; indeed by infinite descent).
* Fermat claimed (Fermat's Last Theorem) to have shown there are no solutions to $x^n + y^n = z^n$ for all $n\geq 3$; this claim appears in his annotations in the margins of his copy of Diophantus.

Euler

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateurUp to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way . (There were already some recognisable features of professional ''practice'', viz., seeking correspondents, visiting foreign colleagues, building private libraries . Matters started to shift in the late seventeenth century ; scientific academies were founded in England (the Royal Society, 1662) and France (the Académie des sciences, 1666) and Russia (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ( and
).
In this context, the term ''amateur'' usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy ; cited in ). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions. Goldbach, pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following:
* ''Proofs for Fermat's statements.'' This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that $p = x^2 + y^2$ if and only if $p\equiv 1 \bmod 4$; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange