research in number theory

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 Bucur

Research Areas

Coxeter Groups

Kac-Moody Algebras

Metaplectic Forms

Automorphic Forms

Daniel Kane

Combinatorics

Kiran Kedlaya

Algebraic Geometry

Arithmetic Algebraic Geometry

Aaron Pollack

Cristian Popescu

Algebraic Number Theory

Arithmetic Geometry

Additional Faculty

Alireza Salehi Golsefidy

Number Theory

Arithmetic lattices

Homogeneous dynamical systems

Claus Sorensen

Langlands program

9500 Gilman Drive, La Jolla, CA 92093-0112

(858) 534-3590

Department of Mathematics

Algebra 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 faculty

At 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, Professor

Jon 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 Emerita

Dr. 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 profile 

Seminars, upcoming conferences, and programs.

• Oregon Number Theory Days - An ongoing series of mini-conferences meeting three times per year, rotating between Oregon State University, the University of Oregon, and Portland State University
• OSU REU Program - For over 30 years OSU has hosted NSF supported summer research opportunities, often in number theory and related areas.

Past conferences

• 25th Automorphic Forms Workshop, March 23-26, 2011, Oregon State University
• Computational Representation Theory in Number Theory, July 27-31 2015, Oregon State University
• 2016 Pacific Northwest Number Theory Conference

Courses taught by faculty in algebra/number theory:

• MTH 343 , Introduction to Modern Algebra
• MTH 440/540 , Computational Number Theory
• MTH 441/541 , Applied and Computational Algebra: Algebraic Coding Theory
• MTH 442/542 , Applied and Computational Algebra: Groebner Bases
• MTH 443/543 , Abstract Linear Algebra
• MTH 444 , Abstract Algebra
• MTH 644 , Abstract Algebra I
• MTH 645 , Abstract Algebra II
• MTH 649 , Topics in Algebra and Number Theory. Recent topics covered under this designation include Commutative Rings and Modules, Algebraic Geometry, Introduction to p-adic Analysis, Ergodic Theory and Translation Surfaces, Arithmetic Dynamics, Algebraic Number Theory, Riemann Surfaces, Class Field Theory, Modular Forms, Hypergeometric Functions.

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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 .

• 1.1.1 Dawn of arithmetic
• 1.1.2 Classical Greece and the early Hellenistic period
• 1.1.3 Diophantus
• 1.1.4 Āryabhaṭa, Brahmagupta, Bhāskara
• 1.1.5 Arithmetic in the Islamic golden age
• 1.1.6 Western Europe in the Middle Ages
• 1.2.1 Fermat
• 1.2.2 Euler
• 1.2.3 Lagrange, Legendre, and Gauss
• 1.3 Maturity and division into subfields
• 2.1 Elementary number theory
• 2.2 Analytic number theory
• 2.3 Algebraic number theory
• 2.4 Diophantine geometry
• 3.1 Probabilistic number theory
• 3.2 Arithmetic combinatorics
• 3.3 Computational number theory
• 4 Licensing

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 period

Aside 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, Bhāskara

While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an 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 Ages

Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica .

Early modern number theory

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:

• One of Fermat's first interests was perfect numbers (which appear in Euclid, Elements IX) and amicable numbers; these topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.
• In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.

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:

• Pell's equation , first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation.
• First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.

• Diophantine equations . Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.

Lagrange, Legendre, and Gauss

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:

• The rise to self-consciousness of number theory (or higher arithmetic ) as a field of study.
• The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.
• The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).

The 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 subdivisions

Elementary 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

• in terms of its tools, as the study of the integers by means of tools from real and complex analysis; or
• in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.

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 theory

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

The 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 subfields

The 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 theory

Much 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 combinatorics

Computational 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:

• Number theory, Wikipedia under a CC BY-SA license

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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 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.

• PhD: Columbia University, 1991
• Interests: Number Theory
• Office: Room 6724
• email: [email protected]
• PhD: McGill University, 2013
• Interests: Number Theory and Arithmetic Geometry
• Office: Room 6511
• email : [email protected]
• PhD:  Columbia University, 2016
• Interests:  Number Theory
• Office:   Room 6512
• email : [email protected]
• PhD: University of California, San Diego, 1986
• Office: Room 6524
• email: [email protected]
• PhD:  Purdue University, 1991
• Interests:  Analytic Number Theory
• Office: Room 6721
• email:  [email protected]

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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. 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 Seminars

One 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 Conferences

Analytic and Combinatorial Number Theory: The Legacy of Ramanujan , June 6-9, 2019

Illinois has hosted a long running series of number theory conferences:

• 2007 Illinois Number Theory Fest
• 2009 Illinois Number Theory Celebration
• 2010 Illinois Number Theory Conference
• 2011 Illinois Number Theory Conference
• 2014 Illinois Number Theory: A Conference in Memory of the Batemans and Heini Halberstam
• 2015 Illinois Number Theory Conference

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 interests

Scott 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 areas

József 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 Theory

To be posted.

Emeritus Faculty in Number Theory

Harold 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:

• Math 453 , Elementary Theory of Numbers (upper undergraduate level)  
• Math 530 , Algebraic Number Theory (beginning graduate course)  
• Math 531 , Analytic Theory of Numbers, I (beginning graduate course)

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:

• Algebraic: Class field theory, elliptic curves, modular forms, Iwasawa theory, algorithmic number theory, cryptography, Abelian varieties.  
• Analytic: Exponential sums, Asymptotic methods in analysis, Riemann zeta-function and L-functions, sieve methods, probabilistic number theory, elliptic functions, Continued fractions, Modular forms, Theory of partitions, additive number theory, Uniform distribution, Distribution of sequences, hypergeometric functions, Diophantine approximation, polynomials, elementary methods, anatomy of integers.

Graduate Awards

The 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 theory

Francesco 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.

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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?

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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.

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