Number theory

For the book by André Weil, see Number Theory: An Approach Through History from Hammurapi to Legendre.

Not to be confused with Numerology.

The distribution of prime numbers, a central point of study in number theory, illustrated by an Ulam spiral. It shows the conditional independence between being prime and being a value of certain quadratic polynomials.

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 $\{1,2,3,\dots \}$ to include number $0$ and the negation of natural numbers $\{-1,-2,-3,\dots \}$. 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 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

Plimpton 322 tablet

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 takiltum 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

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

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. 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

Further information: Ancient Greek mathematics

Although other civilizations probably influenced Greek mathematics at the beginning, all evidence of such borrowings appear relatively late, and it is likely that Greek arithmētikḗ (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 arithmētikḗ and calculation (logistikē). Plato reports in his dialogue Theaetetus that Theodorus had proven that ${\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}$ 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: Platonem ferunt didicisse Pythagorea omnia ("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 [= divides] an even number, then it also measures [= divides] half of it". This is all that is needed to prove that ${\sqrt {2}}$ is irrational. Pythagoreans apparently gave great importance to the odd and the even. The discovery that ${\sqrt {2}}$ is irrational is credited to the early Pythagoreans, sometimes assigned to Hippasus, who was expelled or split from the Pythagorean community as a result. 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

Title page of Diophantus's Arithmetica, translated into Latin by Bachet (1621)

Aside from the elementary work of Neopythagoreans such as Nicomachus and Theon of Smyrna, the foremost authority in arithmētikḗ 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 Arithmetica 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). (There is one important step glossed over in Sunzi's solution: 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 (大衍術) 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, 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; 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

Further information: Mathematics in medieval Islam and Islamic Golden Age

Al-Haytham as seen by the West: on the frontispiece of Selenographia Alhasen [sic] represents knowledge through reason and Galileo knowledge through the senses.

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 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

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; 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 $a^{p-1}\equiv 1{\bmod {p}}.$
• 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 $x^{2}-Ny^{2}=1$ 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 $x^{4}+y^{4}=z^{4}$ 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

Leonhard Euler

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur 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 (1770), soon improved by Euler himself); the lack of non-zero integer solutions to $x^{4}+y^{4}=z^{2}$ (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).
• Pell's equation, first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation.
• First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.
• Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form $x^{2}+Ny^{2}$, some of it prefiguring quadratic reciprocity.
• Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.

Lagrange, Legendre, and Gauss

Carl Friedrich Gauss

Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations; for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to $mX^{2}+nY^{2}$), including defining their equivalence relation, showing how to put them in reduced form, etc.

Adrien-Marie Legendre (1752–1833) was the first to state 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 $ax^{2}+by^{2}+cz^{2}=0$ and worked on quadratic forms along the lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for $n=5$ (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).

Carl Friedrich Gauss (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The Disquisitiones Arithmeticae (1801), which he wrote three years earlier when he was 21, 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 and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (congruences) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory:

The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.

In this way, Gauss arguably made forays towards Évariste Galois's work and the area algebraic number theory.

Maturity and division into subfields

Peter Gustav Lejeune Dirichlet

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.
• 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), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), 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; 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).

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.

Main subdivisions

Elementary number theory

Number theorists Paul Erdős and Terence Tao in 1985, when Erdős was 72 and Tao was 10

Elementary number theory deals with the topics in number theory by means of basic methods in arithmetic. Its primary subjects of study are divisibility, factorization, and primality, as well as congruences in modular arithmetic. Other topics in elementary number theory include Diophantine equations, continued fractions, integer partitions, and Diophantine approximations.

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithms. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed the product, such as $2\times 3=6$.

Divisibility is a property between two nonzero integers related to division. An integer $a$ is said to be divisible by a nonzero integer $b$ if $a$ is a multiple of $b$; that is, if there exists an integer $q$ such that $a=bq$. An equivalent formulation is that $b$ divides $a$ and is denoted by a vertical bar, which in this case is $b|a$. Conversely, if this were not the case, then $a$ would not be divided evenly by $b$, resulting in a remainder. Euclid's division lemma asserts that $a$ and $b$ can generally be written as $a=bq+r$, where the remainder $r<b$ accounts for the leftover quantity. Elementary number theory studies divisibility 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 decimal digit sum is divisible by 3.

A common divisor of several nonzero integers is an integer that divides all of them. The greatest 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. The Euclidean algorithm computes the greatest common divisor of two integers $a,b$ by means of repeatedly applying the division lemma and shifting the divisor and remainder after every step. The algorithm can be extended to solve a special case of linear Diophantine equations $ax+by=1$. A Diophantine equation is an equation with several unknowns and integer coefficients. Another kind of Diophantine equation is described in the Pythagorean theorem, $x^{2}+y^{2}=z^{2}$, whose solutions are called Pythagorean triples if they are all integers.

Elementary number theory studies the divisibility properties of integers such as parity (even and odd numbers), prime numbers, and perfect numbers. Important number-theoric functions include the divisor-counting function, the divisor summatory function and its modifications, and Euler's totient function. A prime 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 {2, 3, 5, 7, 11, ...}. The sieve of Eratosthenes was devised as an efficient algorithm for identifying all primes up to a given natural number by eliminating all composite numbers.

Factorization is a method of expressing a number as a product. 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 as prime factorization. A fundamental property of primes is shown in Euclid'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. The unique 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, $120$ is expressed uniquely as $2\times 2\times 2\times 3\times 5$ or simply $2^{3}\times 3\times 5$.

Modular arithmetic works with finite sets of integers and introduces the concepts of congruence and residue classes. A congruence of two integers $a,b$ modulo $n$ (a positive integer called the modulus) is an equivalence relation whereby $n|(a-b)$ is true. Performing Euclidean division on both $a$ and $n$, and on $b$ and $n$, yields the same remainder. This written as ${\textstyle a\equiv b{\pmod {n}}}$. 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 $n$ is a set that contains all integers congruent to a specified $r$ modulo $n$. For example, $6\mathbb {Z} +1$ 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 is Fermat's little theorem, which states that if a prime $p$ is coprime to some integer $a$, then ${\textstyle a^{p-1}\equiv 1{\pmod {p}}}$ is true. Euler's theorem extends this to assert that every integer $n$ satisfies the congruence$$a^{\varphi (n)}\equiv 1{\pmod {n}},$$where Euler's totient function $\varphi$ counts all positive integers up to $n$ that are coprime to $n$. Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra, such as the Chinese remainder theorem.

Analytic number theory

Main article: Analytic number theory

Riemann zeta function ζ(s) in the complex plane. The color of a point s gives the value of ζ(s): dark colors denote values close to zero and hue gives the value's argument.

The action of the modular group on the upper half plane. The region in grey is the standard fundamental domain.

Analytic number theory, in contrast to elementary number theory, relies on complex 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 from real and complex analysis; 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.

It studies the distribution of primes, behavior of number-theoric functions, and irrational numbers.

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. The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture, the twin prime conjecture, the Hardy–Littlewood conjectures, the Waring problem and the Riemann hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.

Analysis is the branch of mathematics that studies the limit, 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 ${\textstyle {\frac {1}{x}}}$ as $x$ approaches infinity is 0. The complex numbers extend the real numbers with the imaginary unit $i$ defined as the solution to $i^{2}=-1$. Every complex number can be expressed as $x+iy$, where $x$ is called the real part and $y$ is called the imaginary part.

The distribution of primes, described by the function $\pi$ 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, including Euler'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 ${\frac {x}{\log(x)}}$ approximates $\pi (x)$. Another distribution involves an offset logarithmic integral which converges to $\pi (x)$ more quickly.

Corrections to an estimate of the prime-counting function using zeros of the zeta function.

The zeta function has been demonstrated to be connected to the distribution of primes. It is defined as the series$$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots$$that converges if $s$ is greater than 1. Euler demonstrated a link involving the infinite product over all prime numbers, expressed as the identity $$\zeta (s)=\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)^{-1}.$$Riemann extended the definition to a complex variable and conjectured that all nontrivial cases ($0<\Re (s)<1$) where the function returns a zero are those in which the real part of $s$ is equal to ${\textstyle {\frac {1}{2}}}$. He established a connection between the nontrivial zeroes and the prime-counting function. In what is now recognised as the unsolved Riemann hypothesis, a solution to it would imply direct consequences for understanding the distribution of primes.

One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.

Elementary number theory works with elementary proofs, a term that excludes the use of complex numbers but may include basic analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous. For example, proofs based on complex Tauberian theorems, such as Wiener–Ikehara, are often seen as quite enlightening but not elementary despite using Fourier analysis, not complex analysis. Here as elsewhere, an elementary 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. Small sieves, for instance, use little analysis and yet still belong to analytic number theory.

Algebraic number theory

Main article: Algebraic number theory

An algebraic number is any complex number that is a solution to some polynomial equation $f(x)=0$ with rational coefficients; for example, every solution $x$ of $x^{5}+(11/2)x^{3}-7x^{2}+9=0$ is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields.

It could be argued that the simplest kind of number fields, namely quadratic fields, were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones Arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form $a+b{\sqrt {d}}$, where $a$ and $b$ are rational numbers and $d$ is a fixed rational number whose square root is not rational.) For that matter, the eleventh-century chakravala 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, when ideal numbers, the theory of ideals and valuation 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 rationals and ${\sqrt {-5}}$, the number $6$ can be factorised both as $6=2\cdot 3$ and $6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})$; all of $2$, $3$, $1+{\sqrt {-5}}$ and $1-{\sqrt {-5}}$ are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws, that is, generalizations of quadratic reciprocity.

Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K. (For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal(L/K) of L over K is an abelian group—are relatively well understood. Their classification was the object of the programme of class field theory, which was initiated in the late nineteenth century (partly by Kronecker and Eisenstein) and carried out largely in 1900–1950.

An example of an active area of research in algebraic number theory is Iwasawa theory. The Langlands 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.

Diophantine geometry

Main article: Diophantine geometry

The central problem of Diophantine geometry is to determine when a Diophantine 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 a curve, a surface, or some other such object in n-dimensional space. In Diophantine geometry, one asks whether there are any rational points (points all of whose coordinates are rationals) or integral 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 finitely or infinitely many rational points on a given curve or surface.

Consider, for instance, the Pythagorean equation $x^{2}+y^{2}=1$. One would like to know its rational solutions, namely $(x,y)$ such that x and y are both rational. This is the same as asking for all integer solutions to $a^{2}+b^{2}=c^{2}$; any solution to the latter equation gives us a solution $x=a/c$, $y=b/c$ to the former. It is also the same as asking for all points with rational coordinates on the curve described by $x^{2}+y^{2}=1$ (a circle of radius 1 centered on the origin).

Two examples of elliptic 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 $f(x,y)=0$, where $f$ is a polynomial in two variables) depends crucially on the genus of the curve. A major achievement of this approach is Wiles's proof of Fermat's Last Theorem, for which other geometrical notions are just as crucial.

There is also the closely linked area of Diophantine approximations: given a number $x$, 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 $a/q$ (with $\gcd(a,q)=1$) a good approximation to $x$ if $|x-a/q|<{\frac {1}{q^{c}}}$, where $c$ is large. This question is of special interest if $x$ is an algebraic number. If $x$ cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in transcendental number theory: if a number can be approximated better than any algebraic number, then it is a transcendental number. It is by this argument that π and e have been shown to be transcendental.

Diophantine geometry should not be confused with the geometry 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, in Faltings's theorem) rather than to techniques in Diophantine approximations.

Other subfields

Main article: Probabilistic number theory

Probabilistic number theory starts with questions such as the following: Take an integer n 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 will n have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average?

Main articles: Arithmetic combinatorics and Additive number theory

Combinatorics in number theory starts with questions like the following: Does a fairly "thick" infinite set $A$ contain many elements in arithmetic progression: $a$,

$a+b,a+2b,a+3b,\ldots ,a+10b$? Should it be possible to write large integers as sums of elements of $A$?

Main article: Computational number theory

A Lehmer sieve, a primitive digital computer used to find primes and solve simple Diophantine equations

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 for testing primality are now known, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.

Applications

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. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.

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 of public-key cryptography algorithms. Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors. These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom 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". Elementary number theory is taught in discrete mathematics courses for computer scientists. It also has applications to the continuous in numerical analysis.

Number theory has now several modern applications spanning diverse areas such as:

• Computer science: The fast Fourier transform (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.
• Physics: The Riemann hypothesis has connections to the distribution of prime numbers and has been studied for its potential implications in physics.
• Error correction codes: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.
• Communications: The design of cellular telephone networks requires knowledge of the theory of modular forms, which is a part of analytic number theory.
• Study of musical scales: the concept of "equal temperament", which is the basis for most modern Western music, involves dividing the octave into 12 equal parts. This has been studied using number theory and in particular the properties of the 12th root of 2.

See also

• Arithmetic dynamics
• Algebraic function field
• Arithmetic topology
• Finite field
• p-adic number
• List of number theoretic algorithms

References

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Further reading

Two of the most popular introductions to the subject are:

• Hardy, G. H.; Wright, E. M. (2008) [1938]. An introduction to the theory of numbers (rev. by D. R. Heath-Brown and J. H. Silverman, 6th ed.). Oxford University Press. ISBN 978-0-19-921986-5.
• Vinogradov, I. M. (2003) [1954]. Elements of Number Theory (reprint of the 1954 ed.). Mineola, NY: Dover Publications.

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:

• Ivan M. Niven; Herbert S. Zuckerman; Hugh L. Montgomery (2008) [1960]. An introduction to the theory of numbers (reprint of the 5th 1991 ed.). John Wiley & Sons. ISBN 978-81-265-1811-1. Retrieved 2016-02-28.
• Rosen, Kenneth H. (2010). Elementary Number Theory (6th ed.). Pearson Education. ISBN 978-0-321-71775-7. Retrieved 2016-02-28.

Popular choices for a second textbook include:

• Borevich, A. I.; Shafarevich, Igor R. (1966). Number theory. Pure and Applied Mathematics. Vol. 20. Boston, MA: Academic Press. ISBN 978-0-12-117850-5. MR 0195803.
• Serre, Jean-Pierre (1996) [1973]. A course in arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer. ISBN 978-0-387-90040-7.

External links

• Number Theory entry in the Encyclopedia of Mathematics
• Number Theory Web