Number Theory
The integers and prime numbers have fascinated people since ancient times. Recently, the field has seen huge advances. The resolution of Fermat's Last Theorem by Wiles in 1995 touched off a flurry of related activity that continues unabated to the present, such as the recent solution by Khare and Wintenberger of Serre's conjecture on the relationship between mod p Galois representations and modular forms. The Riemann hypothesis, a Clay Millennium Problem, is a part of analytic number theory, which employs analytic methods (calculus and complex analysis) to understand the integers. Recent advances in this area include the Green-Tao proof that prime numbers occur in arbitrarily long arithmetic progressions. The Langlands Program is a broad series of conjectures that connect number theory with representation theory. Number theory has applications in computer science due to connections with cryptography.
The research interests of our group include Galois representations, Shimura varieties, automorphic forms, lattices, algorithmic aspects, rational points on varieties, and the arithmetic of K3 surfaces.
Number Theory at MIT
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Department Members in This Field
- Henry Cohn Discrete Mathematics
- Bjorn Poonen Arithmetic Geometry, Algebraic Number Theory, Rational Points on Varieties, Undecidability
- Andrew Sutherland Computational Number Theory and Arithmetic Geometry
- Zhiwei Yun Representation Theory, Number Theory, Algebraic Geometry
- Wei Zhang Number Theory, Automorphic forms, Arithmetic Geometry
Instructors & Postdocs
- Patrick Bieker Arithmetic Geometry, Langlands Program
- Siyan Daniel Li-Huerta Arithmetic Geometry, Langlands Program
- Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry
- Robin Zhang Number Theory, Automorphic Forms, Diophantine Geometry
Researchers & Visitors
- Edgar Costa Computational Number Theory, Arithmetic Geometry
- David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence
- Samuel Schiavone Computational number theory, arithmetic geometry
- Raymond van Bommel Computational Number Theory, Arithmetic Geometry
Graduate Students*
- Niven Achenjang
- Evan Chen Number theory, combinatorics
- Jia (Jane) Shi
- Vijay Srinivasan
*Only a partial list of graduate students
Number TheoryContemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Insights from ergodic theory have led to dramatic progress in old questions concerning the distribution of primes, geometric representation theory and deformation theory have led to new techniques for constructing Galois representations with prescribed properties, and the study of automorphic forms and special values of L-functions have been revolutionized by developments in both p-adic and arithmetic geometry as well as in pure representation theory. The ideas emerging from the Langlands Program (in its many modern guises) and from the developments that grew out of Wiles’ proof of Fermat’s Last Theorem continue to guide much of the ongoing research on the algebraic and geometric sides of the subject, and in the analytic direction the synthesis of additive combinatorics and harmonic analysis continues to lead to breakthroughs in many directions. In addition to specialized graduate courses, the number theory group has a weekly research seminar with outside speakers from across all areas of number theory. There are also a variety of learning seminars aimed at helping students and postdocs to acquire familiarity with important techniques and results that are generally not available in textbooks (with notes posted for wider dissemination when possible). ![]() Daniel Bump![]() Brian Conrad![]() Jared Duker Lichtman![]() Kannan Soundararajan![]() Richard Taylor![]() Sameera Vemulapalli![]() Research in Number Theory![]() Subject Area and Category
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Leave a commentName * Required Email (will not be published) * Required * Required Cancel The users of Scimago Journal & Country Rank have the possibility to dialogue through comments linked to a specific journal. The purpose is to have a forum in which general doubts about the processes of publication in the journal, experiences and other issues derived from the publication of papers are resolved. For topics on particular articles, maintain the dialogue through the usual channels with your editor. ![]() Follow us on @ScimagoJR Scimago Lab , Copyright 2007-2024. Data Source: Scopus® ![]() Cookie settings Cookie Policy Legal Notice Privacy Policy ![]() Number TheoryThe research of the UCLA Number Theory Group is concerned with the arithmetic of modular forms, automorphic forms, Galois representations, Selmer groups, and classical and p-adic L-functions. The group has 5 permanent faculty: Don Blasius, William Duke, Haruzo Hida, Chandrashekhar Khare, and Romyar Sharifi. Each works broadly in number theory and has settled conjectures and introduced new topics of study. As a case in point, Hida’s theory of ordinary p-adic families, and the theories to which it gave rise constitute an area of research that is broadly studied by mathematicians around the world. His theory also provides a powerful tool for applications to Galois representations and p-adic L-functions. Likewise, Khare and his collaborator Wintenberger proved Serre’s Conjecture on modular forms, which was widely viewed as one of the most important conjectures in the field of arithmetic geometry. Since that proof, the extension of this conjecture to automorphic forms has taken hold and stimulated a great deal of research. Duke is a leading analytic number theorist, known for many basic contributions, including the settling, using modular forms, of a well-known equidistribution problem dating back to Gauss. Blasius has made basic contributions to the special values of L-functions, and pioneered, with J. Rogawski, the use of endoscopy in the study of Galois representations. Sharifi formulated a conjecture which provides a explicit refinement to the Iwasawa main conjecture. This has opened up an exciting new avenue of research concerned with the description of arithmetic objects in terms of higher-dimensional geometry. At any given time, the Number Theory Group has two or more postdocs, and up to 10 graduate students. There is a weekly number theory seminar and typically several ongoing instructional seminars devoted to the study of current research papers or topics, and the presentation of research of group members at all levels. Don BlasiusWilliam duke, chandrashekhar khare, romyar sharifi, visiting faculty, yesim demiroglu, kim tuan do, graduate students, ethan alwaise, rohan joshi, timothy smits, jacob swenberg, frederick vu, emeriti faculty, alfred hales, haruzo hida, murray schacher.
ResearchersPast events, graduate program, theses and projects, undergraduates.
Number TheoryNumber Theory is one of the oldest branches of modern mathematics. It is motivated by the study of properties of integers and solutions to equations in integers. Many of its problems can be stated easily, but often require sophisticated methods from a diverse spectrum of areas in order to study. Its modern formulations are wide reaching and have close ties to algebraic geometry, analysis, and group theory; together with computational aspects. Perhaps due to the fundamental and profound nature of the integers, Number Theory plays a special role in mathematics and applications: two of the Clay Millennium Prize Problems are in Number Theory, and many internet security protocols are based on number theoretic problems. Number Theory is an active area of research for faculty at SFU, and together with faculty at UBC , we form one of the largest communities of Number Theory researchers in North America. Current and Upcoming Events
A complete list of our graduate courses can be found here . Information about applying to our program can be found here . The following is a list of the courses relevant to studies in Number Theory. MATH 724 Applications of Complex Analysis MATH 725 Real Analysis MATH 740 Galois Theory MATH 741 Commutative Algebra and Algebraic Geometry MATH 817 Groups and Rings MATH 818 Algebra and Geometry MATH 842 Algebraic Number Theory MATH 843 Analytic and Diophantine Number Theory ApplicationsMATH 447 Coding Theory MACM 401 Introduction to Computer Algebra MACM 442 Cryptography Current Interest and Reading Courses
Undergraduates interested in learning Number Theory can take MATH 342 Elementary Number Theory, which serves as a general introduction with minimal prerequisites. However, because research in Number Theory requires techniques from many areas, we encourage students interested in continuing in this area to take a broad spectrum of courses from our curriculum. For further guidance, please contact the Undergraduate Advisor . Here is a sample of undergraduate research in Number Theory that has been done recently.
For questions about this page, please contact: [email protected]Research Sub-NavigationNumber theory. For Fall 2023, the Number Theory Seminar is running on Thursdays from 2–3PM (occasionally from 1–2PM). Group overviewNumber theorists study properties of the integers, as well as related generating functions (analytic number theory), arithmetic in generalizations of the integers (algebraic number theory), the distribution of rational and algebraic numbers within the reals (Diophantine approximation), and countless other topics. At UBC, the Number Theory group works on sieve methods and the distribution of primes, Diophantine problems, special values of L-functions, arithmetic dynamics, representations of p-adic groups, non-commutative Iwasawa theory and automorphic forms. For prospective studentsNumber theory is an ancient area of mathematical research. Many problems in number theory are so accessible that they can be easily stated to undergraduates, yet so deep that they have withstood attempts to prove them for centuries or even millennia. Number theory has close connections with many other areas such as algebraic geometry, combinatorics, cryptography and coding theory, harmonic analysis, probability, complex analysis, and representation theory. All these connections are explored in the various courses in number theory that we offer each year.
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![]() Number theory is the study of the integers (e.g. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis. In addition, conjectures in number theory have had an impressive track record of stimulating major advances even outside the subject. For example, attempts to prove “Fermat’s Last Theorem” resulted in the development of large areas of algebra over the course of three centuries, and its recent proof involved a profound unifying force in modern mathematics known as the Langlands program. Our specialties include analytic number theory, the Langlands program, the geometry of locally symmetric spaces, arithmetic geometry and the study of algebraic cycles. ![]() Number theory includes many famous questions, both solved and unsolved. For example, Fermat's Last Theorem (that there are no nontrivial integer solutions to x^n + y^n = z^n, with n > 2) is a famous result in number theory, due to Andrew Wiles. Famous open questions in number theory include the Birch and Swinnterton-Dyer conjecture, the Riemann Hypothesis, and Goldbach's conjecture. Modern number theory uses techniques from and contributes to areas across mathematics, including especially representation theory and algebraic geometry. Number theory also plays an important role in computer science, especially in public-key cryptography. ![]() Alina BucurResearch Areas Coxeter Groups Kac-Moody Algebras Metaplectic Forms Automorphic Forms ![]() Daniel KaneCombinatorics ![]() Kiran KedlayaAlgebraic Geometry Arithmetic Algebraic Geometry ![]() Aaron Pollack![]() Cristian PopescuAlgebraic Number Theory Arithmetic Geometry Additional Faculty![]() Alireza Salehi GolsefidyNumber Theory Arithmetic lattices Homogeneous dynamical systems ![]() Claus SorensenLanglands program ![]() 9500 Gilman Drive, La Jolla, CA 92093-0112 (858) 534-3590 Department of MathematicsAlgebra and number theory. ![]() The center of a sunflower contains a pattern of seeds which radiates outward in spiral-shaped rows. Count the rows. Did you count 21? Or perhaps 34? The answer depends on whether you counted the spirals which curve in the clockwise direction, or in the counterclockwise direction. For all sunflowers, these two numbers are consecutive members of the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,.... Scientists believe that this pattern helps the flower maximize the number of seeds it can pack into a given area, to help its chances of reproductive success. Abstract algebra and number theory are broad areas of mathematics which formalize intuitive notions of symmetry, and which explore properties of integers and other closely related number systems. Modern research in these areas includes the exploration of deep questions of a purely theoretical nature, as well as applications to data transmission, information security, physics, and other areas of science and engineering. Algebra and number theory research facultyAt Oregon State, Jon Kujawa works in representation theory and algebraic Lie theory, particularly where it connects with combinatorics, low-dimensional topology, algebraic geometry, or category theory. Clay Petsche researches connections between number theory and dynamical systems, equidistribution, and height functions over local and global fields. Thomas Schmidt studies number theory, the dynamics of continued fractions, and Fuchsian groups related to translation surfaces. Holly Swisher does research in number theory and combinatorics including modular and mock modular forms, partitions, and hypergeometric series. And Mary E. Flahive's work in number theory is principally in Diophantine approximation, with techniques from the geometry of numbers. ![]() Jonathan Kujawa![]() Department Head, ProfessorJon Kujawa works in representation theory and algebraic Lie theory, particularly where it connects with combinatorics, low-dimensional topology, algebraic geometry, or category theory. View Jon's directory profile![]() Clay Petsche![]() Clayton Petsche's research includes the study of algebraic and arithmetic dynamical systems on varieties and analytic spaces, as well as the theory of height functions over global fields. View Clay's directory profile![]() Thomas Schmidt![]() Schmidt is currently most interested in: natural extensions for continued fractions for various Fuchsian groups (with C. Kraaikamp, and various third co-authors: I. Smeets, H. Nakada, and W. Steiner; with K. Calta; and, most recently with P. Arnoux) and connections between the ergodic theory of billards and 1-forms on algebraic curves and, with the ergodic theory and arithmetic of generalized continued fractions. Recent results include diophantine approximation results for flow directions on translation surfaces (with Y. Bugeaud and P. Hubert, and more recently again with P. Hubert); a new proof with A. Fisher of a beautiful result of Moeckel. View Thomas's directory profile![]() Holly Swisher![]() Professor Swisher is a researcher in the areas of number theory and combinatorics. Her current work is focused on modular forms, partition theory, and hypergeometric series, including the interplay between them. View Holly's directory profile![]() Mary E. Flahive![]() Professor EmeritaDr. Flahive's work in number theory is principally in Diophantine approximation, with techniques from the geometry of numbers. She is also working with colleagues in computer science on projects in algebraic coding theory and also on developing and analyzing interconnection networks. View Mary's directory profileSeminars, upcoming conferences, and programs.
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Number theory studies some of the most basic objects of mathematics: integers and prime numbers. It is a huge subject that makes contact with most areas of modern mathematics, and in fact, enjoys a symbiotic relationship with many. The last fifty years in particular have seen some dramatic progress, including Deligne's proof of the Weil conjectures (giving optimal asymptotics for the number of solutions of polynomial equations over finite fields), Faltings' proof of the Mordell conjecture (establishing finiteness of rational points on hyperbolic curves), Wiles' proof of Fermat's last theorem (which brought a 400-year-long quest to completion) and Zhang's proof of the boundedness of prime gaps (taking a huge step towards the twin prime conjecture). The solutions to these problems rely on techniques from many areas, including algebraic and complex geometry, representation theory and modular forms, differential and algebraic topology, and real and complex analysis. Moreover, grand new vistas (such as the Langlands program) have been uncovered, which will surely keep mathematicians busy for decades. The research interests of our group are diverse and reflect the breadth of the subject. They include arithmetic as well as classical algebraic geometry; automorphic, geometric and p-adic representation theory; Shimura varieties and Galois representations; p-adic Hodge theory; harmonic analysis and analytic number theory; representation stability and commutative algebra; and algorithmic and computational number theory. ![]()
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Number Theory![]() Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of analytical objects (for example, 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, for example, as approximated by the latter (Diophantine approximation). The older term for number theory is arithmetic . By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic , and computer science, as in floating point arithmetic .) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic .
Dawn of arithmetic![]() The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 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 only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of "algebra") was exceptionally well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt. The Pythagorean tradition spoke also 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 century). We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in each. The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (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.) There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Classical Greece and the early Hellenistic periodAside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius, PE X, chapter 4 mentions of Pythagoras: "In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad." Aristotle 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 to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm; Elements , Prop. VII.2) and the first known proof of the infinitude of primes ( Elements , Prop. IX.20). In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed 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 we know, such equations were first successfully treated by the Indian school. It is not known whether Archimedes himself had a method of solution. ![]() Diophantus also studied the equations of some non-rational curves, for which no rational parametrisation is possible. He managed to find some rational points on these curves (elliptic curves, as it happens, in what seems to be their first known occurrence) by means of what amounts to a tangent construction: translated into coordinate geometry (which did not exist in Diophantus's time), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve; that other point is a new rational point. (Diophantus also resorted to what could be called a special case of a secant construction.) While Diophantus was concerned largely with rational solutions, he assumed some results on integer numbers, in particular that every integer is the sum of four squares (though he never stated as much explicitly). Āryabhaṭa, Brahmagupta, BhāskaraWhile Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century. Brahmagupta (628 CE) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century). Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke. Arithmetic in the Islamic golden age![]() In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind , which may 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 – ca. 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 AgesOther 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![]() Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs - he had no models in the area. Over his lifetime, Fermat made the following contributions to the field:
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:
![]() Lagrange, Legendre, and Gauss![]() In his Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss (1777–1855) proved 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 a first foray towards both Évariste Galois's work and algebraic number theory. Maturity and division into subfields![]() Starting early in the nineteenth century, the following developments gradually took place:
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 history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on. Main subdivisionsElementary number theory. The term elementary generally denotes a method that does not use complex 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 (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one. ![]() Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, 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. Analytic number theory![]() Analytic number theory may be defined
Some subjects generally considered to be part of analytic number theory, for example, sieve theory, are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or 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. 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.[81] 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. Algebraic number theoryNumber 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 19th 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 geometryThe central problem of Diophantine geometry is to determine when a Diophantine equation has 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 if there are finitely or infinitely many rational points on a given curve (or surface). 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 , however, is a contemporary term for much the same domain as that covered by the term Diophantine geometry . The term arithmetic geometry is arguably used most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, Faltings's theorem) rather than to techniques in Diophantine approximations. Other subfieldsThe areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the 1930s and 1940s, and computational complexity theory from the 1970s. Probabilistic number theoryMuch of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite. At times, a non-rigorous, probabilistic approach leads to a number of heuristic algorithms and open problems, notably Cramér's conjecture. Arithmetic combinatoricsComputational number theory![]() There are two main questions: "Can we compute this?" and "Can we compute it 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. We now know fast algorithms for testing primality, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring. The difficulty of a computation can be useful: modern protocols for encrypting messages (for example, RSA) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems. Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to Hilbert's 10th problem, that there is no Turing machine which can solve all Diophantine equations. In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (We would necessarily be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.) Content obtained and/or adapted from:
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![]() ![]() Department of Mathematics - UC Santa BarbaraNumber theory. Number theory abounds in problems that are easy to state, yet difficult to solve. An example is "Fermat's Last Theorem," stated by Pierre de Fermat about 350 years ago. Finding a proof of this theorem resisted the efforts of many mathematicians who developed new techniques in number theory, for example with the theory of elliptic curves over finite fields. A proof of Fermat's Last Theorem was finally presented by Andrew Wiles in 1995 in a landmark paper in the Annals of Mathematics. Another famous problem from number theory is the Riemann hypothesis. This problem asks for properties of the Riemann zeta function, a function which plays a fundamental role in the distribution of prime numbers. Although it is over one hundred years old the Riemann hypothesis is still unresolved; in fact, the Clay Mathematics Institute has offered a prize of one million dollars for its solution. Yet another famous open problem from number theory is the Goldbach conjecture which states that every even positive integer is a sum of two primes. Understanding this conjecture requires nothing more than high school mathematics, yet it has resisted the efforts of countless mathematicians.
College of Liberal Arts & Sciences Department of Mathematics
Number TheoryThe Department of Mathematics at the University of Illinois at Urbana-Champaign has long been known for the strength of its program in number theory. The department has a large and distinguished faculty noted for their work in this area, and the graduate program in number theory attracts students from throughout the world. At present over twenty students are writing dissertations in number theory. Each semester upper level graduate courses are offered in a variety of topics in analytic, algebraic, combinatorial, and elementary number theory. Weekly SeminarsOne or two regularly scheduled seminars are held each week, with lectures being given by faculty, graduate students, and visiting scholars. The lectures may be elementary introductions, surveys, or expositions of current research. Number Theory ConferencesAnalytic and Combinatorial Number Theory: The Legacy of Ramanujan , June 6-9, 2019 Illinois has hosted a long running series of number theory conferences:
In addition, every two of three years Illinois hosts the Midwest Graduate Number Theory conference for Graduate Students and recent PhDs , a unique type of conference organized almost entirely by graduate students. Number Theory faculty and their research interestsScott Ahlgren [Homepage] — Modular forms and number theory. Patrick Allen [Homepage] (Adjunct Faculty) — Galois representations and automorphic forms. Bruce Berndt [Homepage] — Ramanujan's notebooks, elliptic functions, theta-functions, q -series, continued fractions, character sums, classical analysis. Florin Boca [Homepage] — Diophantine approximation, spacing statistics. Kevin Ford [Homepage] — Arithmetic functions, probabilistic number theory, Weyl sums, comparative prime number theory, sieve theory, Riemann zeta function. Bruce Reznick [Homepage] — Sums of squares of polynomials, combinatorial number theory. Alexandru Zaharescu [Homepage] — Number theory. Faculty in related areasJózsef Balogh — graph theory, extremal combinatorics, additive combinatorics Nathan Dunfield — 3-dimensional geometry and topology, hyperbolic geometry, geometric group theory, experimental mathematics, connections to number theory. Iwan Duursma [Homepage] — Algebraic curves, algebraic coding theory. Vesna Stojanoska — Homotopy theory and its relations to arithmetic. Postdoctoral Faculty in Number TheoryTo be posted. Emeritus Faculty in Number TheoryHarold G. Diamond [ Homepage ] — Prime number theory, sieves, connections with analysis A.J. Hildebrand [Homepage] — Analytic and probabilistic number theory, asymptotic analysis Leon McCulloh — Algebraic number theory, relative Galois module structure of rings of integers, class groups of integral group rings, Stickelberger relations Ken Stolarsky [Homepage] — Diophantine approximation, special functions, geometry of zeros of polynomials Stephen Ullom [Homepage] — Galois modules, class groups Each year, the following one semester courses are offered:
In addition, at least four topics courses are also offered each year, with student input helping to determine the choice of the topics courses. In recent years, enrollment in these courses has been excellent, typically ranging from 10 to 25. We list below some of the topics courses taught between 2005 and 2013:
Graduate AwardsThe Bateman Prize and the Bateman Fellowship are given annually for outstanding research in number theory. They are named for former Professor Paul T. Bateman, who was on the faculty from 1950 to 1989 and continued being active in the group's activities until shortly before his death in 2012. Bateman served as Department Head for the years 1965-1980. Recipients of the Bateman Prize in Number Theory Recipients of the Bateman Fellowship in Number Theory Recent former postdocs in number theoryFrancesco Cellarosi (Postdoc, 2012-2015) Armin Straub (Postdoc, 2012-2015) Xiannan Li (Postdoc, 2011-2013) Jimmy Tseng (Postdoc, 2011-2012) Youness Lamzouri (Postdoc, 2010-2012) Paul Pollack (NSF Postdoc, 2008-2011) Mathew Rogers (NSF Postdoc, 2008-2011) Andrew Schultz (Postdoc, 2007-2010) Jeremy Rouse (Postdoc, 2007-2010) Sung-Geun Lim (Postdoc 2007-2009) Maria Sabitova (Postdoc, 2006-2009) Nayandeep Deka Baruah (Postdoc 2006-2007) Emre Alkan (Postdoc, 2003-2006) Matt Boylan (Postdoc, 2002-2005) Ae Ja Yee (Postdoc, 2000-2003) ![]()
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number theory , branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural of mathematical pursuits. Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background. Until the mid-20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. The advent of digital computers and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes , testing conjectures, and solving numerical problems once considered out of reach. Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers. From prehistory through Classical Greece![]() The ability to count dates back to prehistoric times. This is evident from archaeological artifacts , such as a 10,000-year-old bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something. Very near the dawn of civilization, people had grasped the idea of “multiplicity” and thereby had taken the first steps toward a study of numbers. It is certain that an understanding of numbers existed in ancient Mesopotamia, Egypt , China, and India, for tablets, papyri, and temple carvings from these early cultures have survived. A Babylonian tablet known as Plimpton 322 (c. 1700 bce ) is a case in point. In modern notation, it displays number triples x , y , and z with the property that x 2 + y 2 = z 2 . One such triple is 2,291, 2,700, and 3,541, where 2,291 2 + 2,700 2 = 3,541 2 . This certainly reveals a degree of number theoretic sophistication in ancient Babylon. Despite such isolated results, a general theory of numbers was nonexistent. For this—as with so much of theoretical mathematics—one must look to the Classical Greeks , whose groundbreaking achievements displayed an odd fusion of the mystical tendencies of the Pythagoreans and the severe logic of Euclid ’s Elements (c. 300 bce ). According to tradition, Pythagoras (c. 580–500 bce ) worked in southern Italy amid devoted followers. His philosophy enshrined number as the unifying concept necessary for understanding everything from planetary motion to musical harmony. Given this viewpoint, it is not surprising that the Pythagoreans attributed quasi-rational properties to certain numbers. For instance, they attached significance to perfect numbers —i.e., those that equal the sum of their proper divisors. Examples are 6 (whose proper divisors 1, 2, and 3 sum to 6) and 28 (1 + 2 + 4 + 7 + 14). The Greek philosopher Nicomachus of Gerasa (flourished c. 100 ce ), writing centuries after Pythagoras but clearly in his philosophical debt, stated that perfect numbers represented “virtues, wealth, moderation, propriety, and beauty.” (Some modern writers label such nonsense numerical theology.) In a similar vein, the Greeks called a pair of integers amicable (“friendly”) if each was the sum of the proper divisors of the other. They knew only a single amicable pair: 220 and 284. One can easily check that the sum of the proper divisors of 284 is 1 + 2 + 4 + 71 + 142 = 220 and the sum of the proper divisors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. For those prone to number mysticism, such a phenomenon must have seemed like magic. arXiv's Accessibility Forum starts next month! Help | Advanced Search Number TheoryAuthors and titles for recent submissions.
See today's new changes Fri, 23 Aug 2024 (showing 8 of 8 entries )Thu, 22 aug 2024 (showing 9 of 9 entries ), wed, 21 aug 2024 (showing first 8 of 11 entries ). ![]() Sara Swann, PolitiFact Sara Swann, PolitiFact Leave your feedback
Fact check: Is the DNC offering free abortions to attendees?This fact check originally appeared on PolitiFact . Reproductive rights took center stage during the Democratic National Convention’s first night in Chicago. But is the DNC offering free abortions and vasectomies to attendees, as some conservative social media users have claimed? WATCH: 2024 Democratic National Convention Night 2 RNC Research, an X account run by the Trump campaign and the Republican National Committee, posted Aug. 18, “Democrats are giving out ‘free abortions and vasectomies’ at their convention.” Other users made similar claims on X. A Planned Parenthood branch is providing free medication abortion, vasectomies and emergency contraception through a mobile health clinic in Chicago that’s running at the same time as the DNC. But the convention is not sponsoring or otherwise connected to these services. Planned Parenthood Great Rivers, which is based in the St. Louis region, said Aug. 14 on X and Aug. 19 in a press release that its mobile health unit would be stationed Aug. 19 and 20 in Chicago’s West Loop neighborhood. Planned Parenthood Great Rivers said Aug. 17 that all of its appointment spots had been filled. LIVE FACT CHECK: Night 3 of the Democratic National Convention The DNC is not being held in the West Loop. The event’s nighttime programming and speeches are at the United Center, a few blocks east of the West Loop. Daytime events are at the McCormick Place Convention Center, a few miles south of the West Loop, according to the DNC’s website . The DNC’s website does not list Planned Parenthood as a partner, sponsor or vendor for the event, nor does it mention this mobile health clinic. Planned Parenthood Great Rivers’ press release lists the Chicago Abortion Fund, a nonprofit group, and the Wieners Circle, a food vendor, as partners. It does not mention the DNC. “Meeting patients where they are by offering the mobile clinic’s services in busy areas is yet another continuation of Planned Parenthood’s unending efforts to improve accessibility and expand services for Illinois residents,” the release said, adding that the mobile clinic would also address “the influx of patients” going to Illinois for care as surrounding states restricted reproductive care. Support Provided By: Learn more Educate your inboxSubscribe to Here’s the Deal, our politics newsletter for analysis you won’t find anywhere else. Thank you. Please check your inbox to confirm. ![]() Revisiting the Effective Number Theory for Imbalanced LearningNew citation alert added. This alert has been successfully added and will be sent to: You will be notified whenever a record that you have chosen has been cited. To manage your alert preferences, click on the button below. New Citation Alert!Please log in to your account Information & ContributorsBibliometrics & citations, view options, recommendations, an effective method for imbalanced time series classification: hybrid sampling. Most traditional supervised classification learning algorithms are ineffective for highly imbalanced time series classification, which has received considerably less attention than imbalanced data problems in data mining and machine learning research. ... Clustering Based Undersampling for Effective Learning from Imbalanced Data: An Iterative ApproachThe class imbalance problem is prevalent in many classification tasks such as disease identification using microarray data, network intrusion detection, and so on. These are tasks in which the class distribution is skewed towards one class, more ... Multiset feature learning for highly imbalanced data classificationWith the expansion of data, increasing imbalanced data has emerged. When the imbalance ratio of data is high, most existing imbalanced learning methods decline in classification performance. To address this problem, a few highly imbalanced learning ... InformationPublished in. IEEE Educational Activities Department United States Publication History
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A peer-reviewed journal that publishes original articles on number theory and arithmetic geometry. Covers traditional and emerging areas, reviews, and has a high impact factor and fast submission to decision time.
Browse the latest articles published in Research in Number Theory, a peer-reviewed journal covering all aspects of number theory. Find out the aims and scope, editorial board, submission guidelines and journal updates of this hybrid journal.
A peer-reviewed mathematics journal covering number theory and arithmetic geometry, published by Springer Science+Business Media. Learn about its history, editors, abstracting and indexing, and external links.
Learn about the number theory research interests and projects of the faculty and students at MIT. Explore topics such as Galois representations, Shimura varieties, automorphic forms, lattices, and more.
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777-1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [ 1] Number theorists study prime numbers as well as the properties of ...
Browse the archives of Research in Number Theory, a peer-reviewed journal that publishes original research in number theory and its applications. Find articles from 2015 to 2024 by volume and issue.
Learn about the various topics and methods in number theory, such as analytic, arithmetic, and p-adic number theory, and their applications to algebraic geometry and L-functions. Explore the research interests and achievements of the senior faculty and the number theory group at Columbia University.
Learn about the faculty, courses, and research seminars in number theory at Stanford. Explore the interactions of number theory with ergodic theory, representation theory, automorphic forms, and more.
Research in Number Theory is a peer-reviewed journal covering number theory and arithmetic geometry. It publishes original articles, reviews and letters, and ranks among the top journals in algebra and number theory according to SJR.
At any given time, the Number Theory Group has two or more postdocs, and up to 10 graduate students. There is a weekly number theory seminar and typically several ongoing instructional seminars devoted to the study of current research papers or topics, and the presentation of research of group members at all levels.
Number Theory is an active area of research for faculty at SFU, and together with faculty at UBC, we form one of the largest communities of Number Theory researchers in North America.
Journal of Number Theory publishes selected research articles on number theory and allied areas. Browse the latest issues, articles, news and special issues on topics such as modular forms, p-adic cohomology and applications of automorphic forms.
Number theory is an ancient area of mathematical research. Many problems in number theory are so accessible that they can be easily stated to undergraduates, yet so deep that they have withstood attempts to prove them for centuries or even millennia. Number theory has close connections with many other areas such as algebraic geometry ...
Number Theory. Number theory is the study of the integers (e.g. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients.
Number theory includes many famous questions, both solved and unsolved. For example, Fermat's Last Theorem (that there are no nontrivial integer solutions to x^n + y^n = z^n, with n > 2) is a famous result in number theory, due to Andrew Wiles. Famous open questions in number theory include the Birch and Swinnterton-Dyer conjecture, the Riemann Hypothesis, and Goldbach's conjecture.
Abstract algebra and number theory are broad areas of mathematics which formalize intuitive notions of symmetry, and which explore properties of integers and other closely related number systems. Modern research in these areas includes the exploration of deep questions of a purely theoretical nature, as well as applications to data transmission, information security, physics, and other areas ...
Number theory studies some of the most basic objects of mathematics: integers and prime numbers. It is a huge subject that makes contact with most areas of modern mathematics, and in fact, enjoys a symbiotic relationship with many. The last fifty years in particular have seen some dramatic progress, including Deligne's proof of the Weil ...
Funding: Research grants from funding agencies (please give the research funder and the grant number) and/or research support (including salaries, equipment, supplies, reimbursement for attending symposia, and other expenses) by organizations that may gain or lose financially through publication of this manuscript.
Number Theory The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
Number Theory Number theory abounds in problems that are easy to state, yet difficult to solve. An example is "Fermat's Last Theorem," stated by Pierre de Fermat about 350 years ago. Finding a proof of this theorem resisted the efforts of many mathematicians who developed new techniques in number theory, for example with the theory of elliptic curves over finite fields. A proof of Fermat's ...
Number Theory. The Department of Mathematics at the University of Illinois at Urbana-Champaign has long been known for the strength of its program in number theory. The department has a large and distinguished faculty noted for their work in this area, and the graduate program in number theory attracts students from throughout the world.
Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, and geometric number theory.
Browse the latest research papers in number theory, a branch of mathematics that studies the properties and structure of integers and related objects. Find titles, authors, abstracts, and links to full texts of 68 entries from August 2024.
RNC Research, an X account run by the Trump campaign and the Republican National Committee, posted Aug. 18, "Democrats are giving out 'free abortions and vasectomies' at their convention."
Historical trends in nursing theory were explored based on the number of articles published each year. ... nursing research, theoretical models, and cancer chronic diseases. From 2001 to 2012, nursing theory research formed fifteen research hotspots, including health promotion, nurse decision-making, evidence-based practice, methodology, long ...
An enhanced effective number theory is established in which data scatter and covering offset among different categories are involved. Subsequently, a new weight calculation manner is proposed based on our new theory, yielding a new loss, namely, NENum loss. In this loss, weights are sample-wise instead of category-wise used in the existing ...
References provide the information necessary for readers to identify and retrieve each work cited in the text. Consistency in reference formatting allows readers to focus on the content of your reference list, discerning both the types of works you consulted and the important reference elements with ease.