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# Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers. 

## Abelian and tauberian theorems

In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber.

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition.

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

## Al-Karaji

(c. 953 &ndash; c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad.

## Al-Ma'mun

Abu al-Abbas al-Maʾmūn ibn Hārūn al-Rashīd (أبو العباس المأمون; September 786 – 9 August 833) was the seventh Abbasid caliph, who reigned from 813 until his death in 833.

## Algebraic curve

In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

## Algebraic function field

In mathematics, an (algebraic) function field of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K.

## Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

## Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

## Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

## Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

## Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two.

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Amicable numbers

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.

## An Introduction to the Theory of Numbers

An Introduction to the Theory of Numbers is a classic book in the field of number theory, by G. H. Hardy and E. M. Wright.

## Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.

## Ancient Egyptian mathematics

Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. 300 BC, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt.

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Archimedes's cattle problem

Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions.

## Argument (complex analysis)

In mathematics, the argument is a multi-valued function operating on the nonzero complex numbers.

## Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

## Arithmetica

Arithmetica (Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD.

## Aryabhata

Aryabhata (IAST) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.

## Atle Selberg

Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory.

## Automorphic form

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group.

## Évariste Galois

Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.

## Babylonian astronomy

The history of astronomy in Mesopotamia, and the world, begins with the Sumerians who developed the earliest writing system—known as cuneiform—around 3500–3200 BC.

## Babylonian mathematics

Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC.

## Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

## Bhāskara II

Bhāskara (also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhaskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer.

## Brahmagupta

Brahmagupta (born, died) was an Indian mathematician and astronomer.

## Brāhmasphuṭasiddhānta

The Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628.

## Brun sieve

In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Chakravala method

The chakravala method (चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation.

## Chinese remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.

## Christian Goldbach

Christian Goldbach (March 18, 1690 &ndash; November 20, 1764) was a German mathematician who also studied law.

## Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields.

## Claude Gaspard Bachet de Méziriac

Claude Gaspard Bachet de Méziriac (October 9, 1581 – February 26, 1638) was a French mathematician, linguist, poet and classics scholar born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy.

## Cole Prize

The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory.

## Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio is a rational number; otherwise a and b are called incommensurable.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## Computability

Computability is the ability to solve a problem in an effective manner.

## Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

## Computer

A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Computer scientist

A computer scientist is a person who has acquired the knowledge of computer science, the study of the theoretical foundations of information and computation and their application.

## Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure.

## Cramér's conjecture

In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## D. C. Heath and Company

D.C. Heath and Company was an American publishing company located at 125 Spring Street in Lexington, Massachusetts, specializing in textbooks.

## Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the rational numbers Q).

## Diophantine approximation

In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers.

## Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).

## Diophantine geometry

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.

## Diophantus

Diophantus of Alexandria (Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 201 and 215; died around 84 years old, probably sometime between AD 285 and 299) was an Alexandrian Hellenistic mathematician, who was the author of a series of books called Arithmetica, many of which are now lost.

## Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is a non-negative integer.

## Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

## Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.

## Distribution (number theory)

In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.

## Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

## Donald Knuth

Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.

## E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

## Egypt

Egypt (مِصر, مَصر, Khēmi), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia by a land bridge formed by the Sinai Peninsula.

## Elementary arithmetic

Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division.

## Elementary proof

In mathematics, an elementary proof is a mathematical proof that only uses basic techniques.

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

## Epigram

An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement.

## Eratosthenes

Eratosthenes of Cyrene (Ἐρατοσθένης ὁ Κυρηναῖος,; –) was a Greek mathematician, geographer, poet, astronomer, and music theorist.

## Ergodic theory

Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.

## Ernst Kummer

Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclid's theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.

## Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

## Eudemus of Rhodes

Eudemus of Rhodes (Εὔδημος) was an ancient Greek philosopher, considered the first historian of science, who lived from c. 370 BC until c. 300 BC.

## Faltings's theorem

In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.

## Fermat Prize

The Fermat prize of mathematical research bi-annually rewards research works in fields where the contributions of Pierre de Fermat have been decisive.

## Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.

## Fermat's little theorem

Fermat's little theorem states that if is a prime number, then for any integer, the number is an integer multiple of.

## Fibonacci

Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".

## Figurate number

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers).

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

## Floating-point arithmetic

In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.

The French Academy of Sciences (French: Académie des sciences) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## G. H. Hardy

Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

## Galois group

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

## Galois theory

In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space.

## Goldbach's conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.

## Gotthold Eisenstein

Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician.

## Gotthold Ephraim Lessing

Gotthold Ephraim Lessing (22 January 1729 – 15 February 1781) was a German writer, philosopher, dramatist, publicist and art critic, and one of the most outstanding representatives of the Enlightenment era.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Hardy–Littlewood circle method

In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory.

## Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

## Harold Davenport

Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.

## Henry Thomas Colebrooke

Henry Thomas Colebrooke FRS FRSE (15 June 1765 – 10 March 1837) was an English orientalist and mathematician.

## Heuristic

A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.

## Hilbert's tenth problem

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900.

## Hippasus

Hippasus of Metapontum (Ἵππασος ὁ Μεταποντῖνος, Híppasos; fl. 5th century BC), was a Pythagorean philosopher.

## Iamblichus

Iamblichus (Ἰάμβλιχος, c. AD 245 – c. 325), was a Syrian Neoplatonist philosopher of Arab origin.

## Ibn al-Haytham

Hasan Ibn al-Haytham (Latinized Alhazen; full name أبو علي، الحسن بن الحسن بن الهيثم) was an Arab mathematician, astronomer, and physicist of the Islamic Golden Age.

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Introduction to Arithmetic

The book Introduction to Arithmetic (Ἀριθμητικὴ εἰσαγωγή, Arithmetike eisagoge) is the only extant work on mathematics by Nicomachus (60&ndash;120 AD).

## Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.

## Jacobi's four-square theorem

Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

## Kuṭṭaka

Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations.

## L-function

In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

## Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Landau prime ideal theorem

In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem.

## Langlands program

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry.

## Large sieve

The large sieve is a method (or family of methods and related ideas) in analytic number theory.

## Larsa

Larsa (Sumerian logogram: UD.UNUGKI, read Larsamki) was an important city of ancient Sumer, the center of the cult of the sun god Utu.

## Leonard Eugene Dickson

Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.

## Magic square

In recreational mathematics and combinatorial design, a magic square is a n\times n square grid (where is the number of cells on each side) filled with distinct positive integers in the range 1,2,...,n^2 such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

## Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

## Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.

There is some confusion in the literature on whether al-Khwārizmī's full name is ابو عبد الله محمد بن موسى الخوارزمي or ابو جعفر محمد بن موسی الخوارزمی.

## Nicomachus

Nicomachus of Gerasa (Νικόμαχος; c. 60 &ndash; c. 120 AD) was an important ancient mathematician best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek.

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Number

A number is a mathematical object used to count, measure and also label.

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## Pappus of Alexandria

Pappus of Alexandria (Πάππος ὁ Ἀλεξανδρεύς; c. 290 &ndash; c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry.

## Paul Erdős

Paul Erdős (Erdős Pál; 26 March 1913 – 20 September 1996) was a Hungarian mathematician.

## Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

## Pearson Education

Pearson Education (see also Pearson PLC) is a British-owned education publishing and assessment service to schools and corporations, as well as directly to students.

## Pell's equation

Pell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x.

## Pentagonal number

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical.

## Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum).

## Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

## Pi

The number is a mathematical constant.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Plato

Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.

## Plimpton 322

Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics.

## Polygonal number

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon.

## Porphyry (philosopher)

Porphyry of Tyre (Πορφύριος, Porphýrios; فرفوريوس, Furfūriyūs; c. 234 – c. 305 AD) was a Neoplatonic philosopher who was born in Tyre, in the Roman Empire.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

## Primality test

A primality test is an algorithm for determining whether an input number is prime.

## Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime number theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

## Probabilistic method

The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object.

## Proof by exhaustion

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.

## Proof by infinite descent

In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle.

## Pure mathematics

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.

## Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

## Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

## Pythagorean triple

A Pythagorean triple consists of three positive integers,, and, such that.

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.

## Qusta ibn Luqa

Qusta ibn Luqa (820–912) (Costa ben Luca, Constabulus) was a Syrian Melkite physician, scientist and translator.

## Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.

## Renaissance

The Renaissance is a period in European history, covering the span between the 14th and 17th centuries.

## Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

## Royal Society

The President, Council and Fellows of the Royal Society of London for Improving Natural Knowledge, commonly known as the Royal Society, is a learned society.

## RSA (cryptosystem)

RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.

The Russian Academy of Sciences (RAS; Росси́йская акаде́мия нау́к (РАН) Rossíiskaya akadémiya naúk) consists of the national academy of Russia; a network of scientific research institutes from across the Russian Federation; and additional scientific and social units such as libraries, publishing units, and hospitals.

## Sieve theory

Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers.

## Solomon Feferman

Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician with works in mathematical logic.

## Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.

## Sophie Germain

Marie-Sophie Germain (1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher.

## Spiral of Theodorus

In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral).

## Square root of 2

The square root of 2, or the (1/2)th power of 2, written in mathematics as or, is the positive algebraic number that, when multiplied by itself, gives the number 2.

## Sunzi Suanjing

Sunzi Suanjing was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty.

## Thales of Miletus

Thales of Miletus (Θαλῆς (ὁ Μιλήσιος), Thalēs; 624 – c. 546 BC) was a pre-Socratic Greek philosopher, mathematician, and astronomer from Miletus in Asia Minor (present-day Milet in Turkey).

## The Princeton Companion to Mathematics

The Princeton Companion to Mathematics is a book, edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, and published in 2008 by Princeton University Press.

## Theaetetus (dialogue)

The Theaetetus (Θεαίτητος) is one of Plato's dialogues concerning the nature of knowledge, written circa 369 BC.

## Theaetetus (mathematician)

Theaetetus of Athens (Θεαίτητος; c. 417 – 369 BC), possibly the son of Euphronius of the Athenian deme Sunium, was a Greek mathematician.

## Theodorus of Cyrene

Theodorus of Cyrene (Θεόδωρος ὁ Κυρηναῖος) was an ancient Libyan Greek and lived during the 5th century BC.

## Thomas Little Heath

Sir Thomas Little Heath (5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## Transcendental number theory

Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with integer coefficients), in both qualitative and quantitative ways.

## Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

## Twin prime

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43).

## Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field.

## Vedas

The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the ''Atharvaveda''. The Vedas (Sanskrit: वेद, "knowledge") are a large body of knowledge texts originating in the ancient Indian subcontinent.

## Waring's problem

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers.

## Wiener–Ikehara theorem

The Wiener–Ikehara theorem is a Tauberian theorem introduced by.

## Wilhelm Xylander

Wilhelm Xylander (born Wilhelm Holtzman, graecized to Xylander; 26 December 153210 February 1576) was a German classical scholar and humanist.

## Wilson's theorem

In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), one has that the factorial (n - 1)!.

## References

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