The distribution ofprime numbers, a central point of study in number theory, illustrated by anUlam spiral. It shows the conditionalindependence between being prime and being a value of certain quadratic polynomials.
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 ofanalytical objects, such as theRiemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also studyreal 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 areFermat's Last Theorem, which was proved 358 years after the original formulation, andGoldbach's conjecture, which remains unsolved since the 18th century. German mathematicianCarl 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 ofpublic-key cryptography algorithms, such as theRSA cryptosystem.
Number theory is the branch of mathematics that studiesintegers and theirproperties and relations.[2] The integers comprise aset that extends the set ofnatural numbers to include number and the negation of natural numbers. Number theorists studyprime numbers as well as the properties ofmathematical 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 thereal 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 termnumber 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]
The Babylonians demonstrated an early understanding of Pythagorean triples.
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 thePlimpton 322, dated c. 1800 BC. It is a broken clay tablet that contains a list ofPythagorean triples, that is, integers such that. The triples are too numerous and too large to have been obtained bybrute force.[20] The table's layout suggests that it was constructed by means of what amounts, in modern language, to theidentity[21]which is implicit in routineOld 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 ofBabylonian 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 Greekarithmētikḗ (the theoretical or philosophical study of numbers) is an indigenous tradition.[28] The ancient Greeks developed a keen interest indivisibility. ThePythagoreans attributed mystical quality toperfect andamicable numbers. The Pythagorean tradition also spoke of so-calledpolygonal orfigurate numbers.[29] Euclid devoted part of hisElements to topics that belong to elementary number theory, includingprime numbers anddivisibility.[30] He gave theEuclidean algorithm for computing thegreatest common divisor of two numbers and aproof implying the infinitude of primes. The foremost authority inarithmētikḗ in Late Antiquity wasDiophantus of Alexandria, who probably lived in the 3rd century AD. He wroteArithmetica, 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 or. In modern parlance,Diophantine equations arepolynomial equations to which rational or integer solutions are sought.
After the fall of Rome, development shifted to Asia, albeit intermittently. TheChinese remainder theorem appears as an exercise[31] inSunzi Suanjing (between the third and fifth centuries).[32] The result was later generalized with a complete solution calledDa-yan-shu (大衍術) inQin Jiushao's 1247Mathematical 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, could be solved by a method he calledkuṭṭaka, orpulveriser;[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, thePell equation. A general procedure for solving Pell's equation was probably found byJayadeva; the earliest surviving exposition appears inBhāskara II's Bīja-gaṇita (twelfth century).[40]
In the early ninth century, the caliphal-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work.[41][42]Diophantus's main work, theArithmetica, was translated into Arabic byQusta ibn Luqa (820–912).Part of the treatiseal-Fakhri (byal-Karajī, 953 – c. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporaryIbn al-Haytham knew[43] what would later be calledWilson's theorem. Other than a treatise on squares in arithmetic progression byFibonacci no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the lateRenaissance, 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]
Fermat is sometimes considered the founder of modern number theory.
The interest ofLeonhard 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, includingFermat'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 betweencontinued fractions and Pell's equation.[55][56] He made the first steps towards analytic number theory.[57]
The Riemann hypothesis is of interest in analytic number theory.
Starting early in the nineteenth century, the following developments gradually took place:
The rise to self-consciousness of number theory (orhigher 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 andcyclotomy, but truly came into its own with the development ofabstract algebra and early ideal theory andvaluation theory; see below. A conventional starting point for analytic number theory isDirichlet's theorem on arithmetic progressions (1837),[64][65] whose proof introducedL-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 ofcomplex analysis in number theory comes later: the work ofBernhard Riemann (1859) on thezeta 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]
Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations ofaddition,subtraction,multiplication,division,exponentiation, extraction ofroots, andlogarithms. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed theproduct, such as.[73]
Divisibility is a property between two nonzero integers related to division. An integer is said to be divisible by a nonzero integer if is a multiple of; that is, if there exists an integer such that. An equivalent formulation is that divides and is denoted by a vertical bar, which in this case is. Conversely, if this were not the case, then would not be divided evenly by, resulting in a remainder.Euclid's division lemma asserts that and can generally be written as, where the remainder accounts for the smallest positive leftover quantity. Elementary number theory studiesdivisibility rules in order to quickly identify if a given integer is divisible by a fixed divisor. For instance, it is known that any integer is divisible by 3 if its decimaldigit sum is divisible by 3.[74][9][75]
Example of a continued fraction.
A common divisor of several nonzero integers is an integer that divides all of them. Thegreatest common divisor (gcd) is the largest of such divisors. Two integers are said to be coprime or relatively prime to one another if their greatest common divisor, and simultaneously their only divisor, is 1. TheEuclidean algorithm computes the greatest common divisor of two integers by means of repeatedly applying the division lemma and shifting the divisor and remainder after every step. The algorithmcan be extended to solve a special case oflinear Diophantine equations. A Diophantine equation has several unknowns and integer coefficients. Another kind of Diophantine equation is described in thePythagorean theorem,, whose solutions are called Pythagorean triples if they are all integers.[9][10] Another kind of expression is thecontinued fraction, which writes a sum of an integer and a fraction whose denominator is another such sum.[76]
Elementary number theory studies the divisibility properties of integers such asparity (even and odd numbers),prime numbers, andperfect numbers. Important number-theoretic functions include thedivisor-counting function, thedivisor summatory function and its modifications, andEuler's totient function. Aprime number is an integer greater than 1 whose only positive divisors are 1 and the prime itself. A positive integer greater than 1 that is not prime is called a composite number.Euclid's theorem demonstrates that there are infinitely many prime numbers that comprise the set. Thesieve of Eratosthenes was devised as an efficient algorithm for identifying all primes up to a given natural number by eliminating all composite numbers.[77]
Factorization is a method of expressing a number as aproduct. Specifically in number theory,integer factorization is the decomposition of an integer into a product of integers. The process of repeatedly applying this procedure until all factors are prime is known asprime factorization. A fundamental property of primes is shown inEuclid's lemma. It is a consequence of the lemma that if a prime divides a product of integers, then that prime divides at least one of the factors in the product. Theunique factorization theorem is the fundamental theorem of arithmetic that relates to prime factorization. The theorem states that every integer greater than 1 can be factorised into a product of prime numbers and that this factorisation is unique up to the order of the factors. For example, is expressed uniquely as or simply.[78][9]
Modular arithmetic works with finite sets of integers and introduces the concepts of congruence and residue classes. A congruence of two integers modulo (a positive integer called the modulus) is anequivalence relation whereby is true. PerformingEuclidean division on both and, and on and, yields the same remainder. This written as. In a manner analogous to the 12-hour clock, the sum of 4 and 9 is equal to 13, yet congruent to 1. A residue class modulo is a set that contains all integers congruent to a specified modulo. For example, contains all multiples of 6 incremented by 1. Modular arithmetic provides a range of formulas for rapidly solving congruences of very large powers. An influential theorem isFermat's little theorem, which states that if a prime is coprime to some integer, then is true.Euler's theorem extends this to assert that every integer satisfies the congruencewhere Euler's totient function counts all positive integers up to that are coprime to. Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra, such as theChinese remainder theorem.[79]
Analytic number theory, in contrast to elementary number theory, relies oncomplex numbers and techniques from analysis and calculus. Analytic number theory may be defined
in terms of its tools, as the study of the integers by means of tools fromreal andcomplex analysis;[64] or
in terms of its concerns, as the study within number theory of estimates on the size and density of certain numbers (e.g., primes), as opposed to identities.[80]
It studies the distribution of primes, behavior of number-theoretic functions, and irrational numbers.[81]
Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, many of the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.[82] The following are examples of problems in analytic number theory: theprime number theorem, theGoldbach conjecture, thetwin prime conjecture, theHardy–Littlewood conjectures, theWaring problem and theRiemann hypothesis. Some of the most important tools of analytic number theory are thecircle method,sieve methods andL-functions (or, rather, the study of their properties). The theory ofmodular forms (and, more generally,automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.[83]
Analysis is the branch of mathematics that studies thelimit, defined as the value to which a sequence or function tends as the argument (or index) approaches a specific value. For example, the limit of the sequence 0.9, 0.99, 0.999, ... is 1. In the context of functions, the limit of as approaches infinity is 0.[84] The complex numbers extend the real numbers with the imaginary unit defined as the solution to. Every complex number can be expressed as, where is called the real part and is called the imaginary part.[85]
Thedistribution of primes, described by the function that counts all primes up to a given real number, is unpredictable and is a major subject of study in number theory. Elementary formulas for a partial sequence of primes, includingEuler's prime-generating polynomials have been developed. However, these cease to function as the primes become too large. The prime number theorem in analytic number theory provides a formalisation of the notion that prime numbers appear less commonly as their numerical value increases. One distribution states, informally, that the function approximates. Another distribution involves an offset logarithmic integral which converges to more quickly.[3]
Corrections to anestimate of the prime-counting function using zeros of the zeta function
Thezeta function has been demonstrated to be connected to the distribution of primes. It is defined as the seriesthat converges if is greater than 1. Euler demonstrated a link involving the infinite product over all prime numbers, expressed as the identityRiemann extended the definition to a complex variable and conjectured that all nontrivial cases () where the function returns a zero are those in which the real part of is equal to. He established a connection between the nontrivial zeroes and the prime-counting function. In what is now recognised as the unsolvedRiemann hypothesis, a solution to it would imply direct consequences for understanding the distribution of primes.[86]
One may ask analytic questions aboutalgebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may defineprime ideals (generalizations ofprime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This questioncan be answered by means of an examination ofDedekind zeta functions, which are generalizations of theRiemann zeta function, a key analytic object at the roots of the subject.[87] This is an example of a general procedure in analytic number theory: deriving information about the distribution of asequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.[88]
Elementary number theory works withelementary proofs, a term that excludes the use ofcomplex numbers but may include basic analysis.[72] For example, theprime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 byErdős andSelberg.[89] The term is somewhat ambiguous. For example, proofs based on complexTauberian theorems, such asWiener–Ikehara, are often seen as quite enlightening but not elementary despite usingFourier analysis, not complex analysis. Here as elsewhere, anelementary proof may be longer and more difficult for most readers than a more advanced proof.
Some subjects generally considered to be part of analytic number theory (e.g.,sieve theory) are better covered by the second rather than the first definition.[note 5] Small sieves, for instance, use little analysis and yet still belong to analytic number theory.[note 6]
Analgebraic number is anycomplex number that is a solution to some polynomial equation with rational coefficients; for example, every solution of is an algebraic number. Fields of algebraic numbers are also calledalgebraic number fields, or shortlynumber fields. Algebraic number theory studies algebraic number fields.[90]
It could be argued that the simplest kind of number fields, namelyquadratic fields, were already studied by Gauss, as the discussion of quadratic forms inDisquisitiones Arithmeticae can be restated in terms ofideals andnorms in quadratic fields. (Aquadratic field consists of allnumbers of the form, where and are rational numbers andis a fixed rational number whose square root is not rational.)For that matter, the eleventh-centurychakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.
The grounds of the subject were set in the late nineteenth century, whenideal numbers, thetheory of ideals andvaluation theory were introduced; these are three complementary ways of dealing with the lack of unique factorization in algebraic number fields. (For example, in the field generated by the rationalsand, the number can be factorised both as and; all of,, andare irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (byKummer) seems to have come from the study of higher reciprocity laws,[91] that is, generalizations ofquadratic reciprocity.
Number fields are often studied as extensions of smaller number fields: a fieldL is said to be anextension of a fieldK ifL containsK.(For example, the complex numbersC are an extension of the realsR, and the realsR are an extension of the rationalsQ.)Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensionsL ofK such that theGalois group[note 7] Gal(L/K) ofL overK is anabelian group—are relatively well understood.Their classification was the object of the programme ofclass field theory, which was initiated in the late nineteenth century (partly byKronecker andEisenstein) and carried out largely in 1900–1950.
An example of an active area of research in algebraic number theory isIwasawa theory. TheLanglands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.
The central problem of Diophantine geometry is to determine when aDiophantine equation has integer or rational solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.
For example, an equation in two variables defines a curve in the plane. More generally, an equation or system of equations in two or more variables defines acurve, asurface, or some other such object inn-dimensional space. In Diophantine geometry, one asks whether there are anyrational points (points all of whose coordinates are rationals) orintegral points (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is whether there are finitelyor infinitely many rational points on a given curve or surface.
Consider, for instance, thePythagorean equation. One would like to know its rational solutions, namely such thatx andy are both rational. This is the same as asking for all integer solutionsto; any solution to the latter equation gives us a solution, to the former. It is also thesame as asking for all points with rational coordinates on the curve described by (a circle of radius 1 centered on the origin).
Two examples ofelliptic curves, that is, curves of genus 1 having at least one rational point
The rephrasing of questions on equations in terms of points on curves is felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve (that is, rational or integer solutions to an equation, where is a polynomial in two variables) depends crucially on thegenus of the curve.[note 8] A major achievement of this approach isWiles's proof of Fermat's Last Theorem, for which other geometrical notions are just as crucial.
There is also the closely linked area ofDiophantine approximations: given a number, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call (with) a good approximation to if, where is large. This question is of special interest if is an algebraic number. If cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that ofheight) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest intranscendental number theory: if a number can be approximated better than any algebraic number, then it is atranscendental number. It is by this argument thatπ ande have been shown to be transcendental.
Diophantine geometry should not be confused with thegeometry of numbers, which is a collection of graphical methods for answering certain questions in algebraic number theory.Arithmetic geometry is a contemporary term for the same domain covered by Diophantine geometry, particularly when one wishes to emphasize the connections to modern algebraic geometry (for example, inFaltings's theorem) rather than to techniques in Diophantine approximations.
Probabilistic number theory starts with questions such as the following: Take an integern at random between one and a million. How likely is it to be prime? (this is just another way of asking how many primes there are between one and a million). How many prime divisors willn have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average?
Combinatorics in number theory starts with questions like the following: Does a fairly "thick"infinite set contain many elements in arithmetic progression:,
? Should it be possible to write large integers as sums of elements of?
There are two main questions: "Can this be computed?" and "Can it be computed rapidly?" Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. Fast algorithms fortesting primality are now known, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.
For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics other than the use of prime numbered gear teeth to distribute wear evenly.[92] In particular, number theorists such asBritish mathematicianG. H. Hardy prided themselves on doing work that had absolutely no military significance.[93] The number-theoristLeonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.[94]
This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation ofpublic-key cryptography algorithms.[95] Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.[96] These applications have led to significant study ofalgorithms for computing with prime numbers, and in particular ofprimality testing, methods for determining whether a given number is prime. Prime numbers are also used in computing forchecksums,hash tables, andpseudorandom number generators.
In 1974,Donald Knuth said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".[97]Elementary number theory is taught indiscrete mathematics courses forcomputer scientists. It also has applications to the continuous innumerical analysis.[98]
Number theory has now several modern applications spanning diverse areas such as:
Computer science: Thefast Fourier transform (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.[99]
Physics: TheRiemann hypothesis has connections to the distribution of prime numbers and has been studied for its potential implications in physics.[100]
Error correction codes: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.[101]
Study of musical scales: the concept of "equal temperament", which is the basis for most modern Western music, involves dividing theoctave into 12 equal parts.[102] This has been studied using number theory and in particular the properties of the 12th root of 2.
^The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example,Serre 1996. In 1952,Davenport still had to specify that he meantThe Higher Arithmetic.Hardy and Wright wrote in the introduction toAn Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] toAn 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." (Hardy & Wright 2008)
^Robson 2001, p. 201. This is controversial. SeePlimpton 322. Robson's article is written polemically (Robson 2001, p. 202) with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" (Robson 2001, p. 167); 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 2001, p. 202).
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".(Robson 2001, pp. 199–200)
^See, for example,Sunzi Suanjing, Ch. 3, Problem 36, inLam & Ang 2004, pp. 223–224:
[36] Now 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 [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female.
This is the last problem in Sunzi's otherwise matter-of-fact treatise.
^Up 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 (Weil 1984, pp. 159, 161). (There were already some recognisable features of professionalpractice, viz., seeking correspondents, visiting foreign colleagues, building private libraries (Weil 1984, pp. 160–161). Matters started to shift in the late seventeenth century (Weil 1984, p. 161); scientific academies were founded in England (theRoyal Society, 1662) and France (theAcadémie des sciences, 1666) andRussia (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 (Weil 1984, p. 163 andVaradarajan 2006, p. 7).In this context, the termamateur 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 (Truesdell 1984, p. xv); cited inVaradarajan 2006, p. 9). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.
^The Galois group of an extensionL/K consists of the operations (isomorphisms) that send elements of L to other elements of L while leaving all elements of K fixed.Thus, for instance,Gal(C/R) consists of two elements: the identity element(taking every elementx + iy ofC to itself) and complex conjugation(the map taking each elementx + iy tox − iy).The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started withÉvariste Galois; in modern language, the main outcome of his work is that an equationf(x) = 0 can be solved by radicals(that is,x can be expressed in terms of the four basic operations togetherwith square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equationf(x) = 0 has a Galois group that issolvablein the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)
^Thegenus can be defined as follows: allow the variables in to be complex numbers; then defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of doughnut-like holes in the surface is called thegenus of the curve of equation.
^abKaratsuba, A.A. (2020)."Number theory".Encyclopedia of Mathematics. Springer. Retrieved2025-05-03.
^abMoore, Patrick (2004). "Number theory". In Lerner, K. Lee; Lerner, Brenda Wilmoth (eds.).The Gale Encyclopedia of Science. Vol. 4 (3rd ed.). Gale.ISBN0-7876-7559-8.
^abTanton, James (2005). "Number theory".Encyclopedia of Mathematics. New York: Facts On File. pp. 359–60.ISBN0-8160-5124-0.
^Neugebauer & Sachs 1945, p. 40. The termtakiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Robson 2001, p. 192
^Robson 2001, p. 189. Other sources give the modern formula. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.(van der Waerden 1961, p. 79)
^Neugebauer (Neugebauer 1969, pp. 36–40) discusses the table in detail and mentions in passing Euclid's method in modern notation (Neugebauer 1969, p. 39).
^Herodotus (II. 81) and Isocrates (Busiris 28), cited in:Huffman 2011. On Thales, see Eudemus ap. Proclus, 65.7, (for example,Morrow 1992, p. 52) cited in:O'Grady 2004, p. 1. Proclus was using a work byEudemus of Rhodes (now lost), theCatalogue of Geometers. See also introduction,Morrow 1992, p. xxx on Proclus's reliability.
^Corry, Leo (2015). "Construction Problems and Numerical Problems in the Greek Mathematical Tradition".A Brief History of Numbers. Oxford University Press.ISBN978-0-19-870259-7.
^Sunzi Suanjing, Chapter 3, Problem 26. This can be found inLam & Ang 2004, pp. 219–220, which contains a full translation of theSuan Ching (based onQian 1963). See also the discussion inLam & Ang 2004, pp. 138–140.
^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). SeeLam & Ang 2004, pp. 27–28.
^Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in:Plofker 2008, pp. 134–140. See alsoClark 1930, pp. 42–50. A slightly more explicit description of the kuṭṭaka was later given inBrahmagupta,Brāhmasphuṭasiddhānta, XVIII, 3–5 (inColebrooke 1817, p. 325, cited inClark 1930, p. 42).
^See the discussion in section 5 ofGoldstein & Schappacher 2007. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted inWeil 1984, p. 25).
^Ore, Oystein (1948).Number Theory and Its History (1st ed.). McGraw-Hill.
^Watkins, John J. (2014). "Divisibility".Number Theory: A Historical Approach. Princeton University Press. pp. 76–80.ISBN978-0-691-15940-9.
^Nathanson, Melvyn B. (2000). "Divisibility and Primes".Elementary Methods in Number Theory. Springer.ISBN0-387-98912-9.
^Tanton, James (2005). "Fundamental theorem of arithmetic".Encyclopedia of Mathematics. New York: Facts On File.ISBN0-8160-5124-0.
^Shoup, Victor (2005).A Computational Introduction to Number Theory and Algebra. Cambridge University Press.ISBN978-0-511-11363-5.
^Granville 2008, section 1: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks forgood approximations."
Dauben, Joseph W. (2007), "Chapter 3: Chinese Mathematics", in Katz, Victor J. (ed.),The Mathematics of Egypt, Mesopotamia, China, India and Islam : A Sourcebook, Princeton University Press, pp. 187–384,ISBN978-0-691-11485-9
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Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (Apostol 1981).Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are:
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