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11: Number theory![[The Mathematical Atlas]](http://www.math.niu.edu/~rusin/known-math/index/11-XX.html/../images/title.gif) [Search][Subject Index][MathMap][Tour][Help!]![[MathMap Icon]](http://www.math.niu.edu/~rusin/known-math/index/11-XX.html/../images/mini.gif) ABOUT:[Introduction][History][Related areas][Subfields]POINTERS:[Texts][Software][Web links][Selected topics here]11: Number theoryIntroductionNumber theory is one of the oldest branches of pure mathematics,and one of the largest. Of course, it concerns questions about numbers,usually meaning whole numbers or rational numbers (fractions).Elementary number theory involves divisibility among integers -- thedivision "algorithm", the Euclidean algorithm (and thus the existence ofgreatest common divisors), elementary properties of primes (theunique factorization theorem, the infinitude of primes), congruences(and the structure of the sets Z/nZ as commutative rings), includingFermat's little theorem and Euler's theorem extending it. But theterm "elementary" is usually used in this setting only to mean thatno advanced tools from other areas are used -- not that the resultsthemselves are simple. Indeed, a course in "elementary" number theoryusually includes classic and elegant results such as Quadratic Reciprocity;counting results using the Möbius Inversion Formula (and other multiplicativenumber-theoretic functions); and even the Prime Number Theorem, assertingthe approximate density of primes among the integers, which hasdifficult but "elementary" proofs. Other topics in elementary number theory --the solutions of sets of linear congruence equations (the Chinese RemainderTheorem), or solutions of single binary quadratic equations (Pell's equationsand continued fractions), or the generation of Fibonacci numbers orPythagorean triples -- turn out in retrospect to be harbingers of sophisticatedtools and themes in other areas.The remaining parts of number theory are more or less closely allied withother branches of mathematics, and typically use tools from those areas.For example, many questions in number theory may be posed asDiophantine equations -- equations to be solved in integers --without much preparation. Catalan's conjecture -- are 8 and 9 the only consecutive powers? -- asks for the solution to xa-yb=1 in integers; the Four Squares Theorem -- every natural number is the sum offour integer squares -- simply asserts that x² + y² + z² + w² = n is solvable for all n. Butthe attempt to solve these equations requires rather powerful tools fromelsewhere in mathematics to shed light on the the structure of the problem.(Even the possibility of analyzing Diophantine equations -- Hilbert'stenth problem -- suggests the use of mathematical logic; Matijasevic'snegative solution of that problem guarantees number theorists will neverfind a complete solution to their analyses!)We can try to subdivide number theory according to those other tools used.Naturally there is significant overlap, and a single question from elementarynumber theory often requires tools from many branches of number theory."Combinatorial Number Theory" involves the number-theoretic studyof objects which arise naturally from counting or iteration. Thisincludes a study of many specific families of numbers -- the binomialcoefficients, the Fibonacci numbers, Bernoulli numbers, factorials,perfect squares, partition numbers and so on -- which can be obtained by simplerecurrence relations, say, or as values of polynomials. One asks fortheir factorizations, their congruence properties, their densities, etc.It is very easy to state conjectures in this area which can often beunderstood without any particular mathematical training, but which canbe very difficult to solve; Erdös has left many conjectures of this sort."Algebraic Number Theory" extends the concept of "number" to mean anelement of some ring, usually the ring of integers in a finite algebraicextension of the rational number field. These arise naturally even whenconsidering elementary topics (e.g. the representation of an integer asa sum of two squares is tantamount to its factorization in the ring Z[i] of Gaussian integers) but are also interesting in their own right.In this setting, the familiar features of the natural numbers (e.g.unique factorization) need not hold. The virtue of the machineryintroduced -- class groups, discriminants, Galois theory, field cohomology,class field theory, group representations and L-functions -- is that itallows a reconstruction of some of that order in these new settings.A key feature of some problems in number theory is the extent to whichthe behaviour of the problem in integers is reflected in its behaviour modulop for all primes p, and its behaviour in the real line. The correctconstruction for the investigation of this phenomenon is usually alocal ring such as the p-adic integers. These fields provide an opportunityfor unusual forms of analysis (e.g. series converge iff their termsconverge to zero -- the calculus student's dream!) Local analysis usually arises as a part of algebraic number theory."Analytic Number Theory" involves the study of the Riemann zeta functionand other similar functions such as Dirichlet series. The zeta function maybe defined on half the complex plane as the sum 1 + 1/2s + 1/3s + 1/4s + ...;its connection with number theory results from its factorization asa product Prod(1 - 1/p^s )^(-1), the product taken over all primes p.Thus for example the distribution of the primes among the integers can bededuced from a good understanding of the behaviour of zeta(s). The RiemannHypothesis states that zeta(s) is never zero except along the lineRe(s)=1/2 (or at the negative even integers). This is arguably the most important open question in mathematics. There are other related functions, useful either for studying the Riemann zeta function or for making similar conclusions about other sets; for example, one may use them to prove theinfinitude of primes in candidate linear progressions.Other areas of number theory are also quite analytical. For example,"additive number theory" asks about ways of expressing an integer Nas a sum of integers a_i in a set A. If we set f(z) = Sum exp(2 pi i a_i z),then f(z)^k has exp(2 pi i N z) as a summand iff N is a sum of k ofthe a_i. This in turn can be deduced from some integration (the integralof exp(a z) around a circle is 0 or not depending on whether a is zero).Thus analytical techniques are used to approach Waring's problem, forexample (representing integers as sums of squares, cubes, etc.), and toaddress other questions with exponential sums. Since these computationsare equivalent to work in the rings Z/nZ, there is an interest in thestructure of these rings. One may include in this analytic category the parts of number theoryconnected with forms (e.g. quadratic forms are quadratic polynomials inseveral variables). Broadly speaking the goal here is to analyze thepossible equivalence classes of functions under groups of symmetries.Even when few analytic tools are used for the analysis of the functionsthemselves, the groups of interest (e.g. the discontinuous groups actingon the complex upper half-plane) are well understood in areas of analysis.Also, ideas from analysis (measure theory, dimension) can be usedin "probabilistic number theory", in which one studiesalmost-periodic, pseudo-random, or fractal behaviour ofnumber-theoretic functions.Finally, a significant amount of analysis is also used in Sieve methods,and other aspects of multiplicative number theory. Here one generalizes thesieve of Eratosthenes to investigate the presence of, say, prime pairs(Brun's sieve) or solutions to the Goldbach conjecture (every even numberis a sum of two primes)."Transcendental number theory" considers proofs of transcendence oralgebraicity of numbers, and the extent to which numbers can beapproximated by algebraic numbers (say). This has a direct bearing onother fields such as Diophantine equations, for example, since theunsolvability of a Diophantine equation can be deduced from theobservation that it would require rational numbers which approximatea real number "too well". Well-known results in this area include the transcendence of pi, which in turn shows the impossibility of squaringthe circle."Geometric number theory" incorporates all forms of geometry. Theclassical Geometry of Numbers due to Minkowski begins with statements ofEuclidean geometry on lattices (A convex body contains a lattice pointif its volume is large enough); by extension this becomes the study ofquadratic forms on lattices, and thus a method of investigatingregular packings of spheres, say. But one may also investigate algebraicgeometry with number theory, that is, one may study varieties such asalgebraic curves and surfaces and ask if they have rational orintegral solutions (points with rational or integral coordinates). Thistopic includes the highly successful theory of elliptic curves (where therational points form a finitely generated group) and finiteness results(e.g. Siegel's, Thue's, or Faltings's) which apply to integral orhigher-genus situations."Computational number theory" studies the effectiveness ofalgorithms for computation of number-theoreticquantities. Considerable effort has been expended in primality-testingand integer factorization routines, for example -- procedures whichare in principle trivial, but whose naive solution is untenable inlarge cases. This field also considers integer quantities (e.g theclass number) whose usual definition is nonconstructive, and realquantities (e.g. the values of zeta functions) which must be computedwith very high precision; thus this overlaps both computer algebra andnumerical analysis.HistoryFairly comprehensive accounts may be found inWeil, André: "Number theory, An approach through history", Birkhäuser Boston, Inc., Boston, Mass., 1984 ISBN 0-8176-3141-0Ore, Oystein, "Number theory and its history", Dover Publications, Inc., New York, 1988. 370 pp. ISBN 0-486-65620-9Dickson, Leonard Eugene: "History of the theory of Numbers", Carnegie Institution of Washington publication. no. 256, 1919: a three-volume history and literature review through early 20th centuryNote that there are many popular books addressing particular aspectsof number theory likely to be of wide interest. Some of these arementioned on index pages for particular subdisciplines. Also likely tobe helpful are biographies of mathematicians who have been prominentacross much of number theory (especially recent romantic figures suchas Ramanujan and Erdös).Applications and related fieldsClearly the separate parts of number theory overlap with other areasof mathematics."Combinatorial number theory", of course, overlaps quite a bit with05: Combinatorics. While here we considernumber-theoretic topics involving the binomial coefficients,partitions, and so on, in combinatorics one might be interested in thesenumbers as real-number sequences, or as tools for counting sets of some type.Questions in algebraic number theory often require tools of Galoistheory; that material is mostly a part of 12: Field theory(particularly the subject of field extensions).The algebraic structure of the ring of integers is similar to thatof other commutative rings such as rings ofpolynomials. (Most material on polynomials is on the Galois Theory page.)The sets of solutions in rational numbers to algebraic equationsmay be viewed as algebraic varieties, and thus studied with tools of14: Algebraic Geometry. This is particularly truewith single equations in two variables (which lead to curves); suchequations when of degree 3 (or 4) lead to elliptic curves.The theory of lattices is arguably one of quadratic forms, which areconsidered in more detail within Linear Algebra.(Note that these are essentially unrelated to the lattices studied alongwith Ordered Sets.)Sequences and series of integers may be viewed asseries and sequences of real numbers, and hencestudied with tools of analysis.Lattices can be used to determine patterns for sphere-packing; consult the sphere FAQ.There is current interest in factoring large integers (and primality testing), in part because of relationships with 94: cryptography.The discipline of computing numerical answers to problems (e.g. locating theroots of a polynomial) is not number theory at all but Numerical Analysis.![[Schematic of subareas and related areas]](http://www.math.niu.edu/~rusin/known-math/index/11-XX.html/../images/11b.gif) Other fields with some overlap as seen in the diagram are areas20 (Group Theory), 81 (Quantum Theory), 22 (Topological groups), 58 (Global Analysis), 01 (History), 68 (Computer Science), 52 (Convex Geometry), 65 (Numerical Analysis), 03 (Logic), 33 (Special Functions). The diagram manages to showthat some parts of number theory are related to fields of algebra (in red),while others involve more discrete mathematics (in purple) or analysis (green).Subfields11A: Elementary number theory, including continued fractions. For analogues in number fields, see 11R0411B: Sequences and sets (including Fibonacci,...)11C: Polynomials and matrices11D: Diophantine equations. Includes Fermat's Last Theorem 11E: Forms and linear algebraic groups See also 19GXX. For quadratic forms in linear algebra, See 15A6311F: Discontinuous groups and automorphic forms See also 11R39,11S37, 14-XX, 22EXX, 14GXX, 14KXX, 22E50, 22E55, 30F35, 32NXX; for relations with quadratic forms, See 11E4511G: Arithmetic algebraic geometry (Diophantine geometry), see also 11DXX, 14-XX, 14GXX, 14KXX11H: Geometry of numbers For applications in coding theory, see 94B7511J: Diophantine approximation, transcendental number theory (including Pi). See also 11K6011K: Probabilistic theory: distribution modulo 1; metric theory of algorithms11L: Exponential sums and character sums (For finite fields, see 11T)11M: Zeta and L-functions: analytic theory11N: Multiplicative number theory11P: Additive number theory; partitions (e.g. Waring's problem)11R: Algebraic number theory: global fields. For complex multiplication, see 11G15. For Galois theory see also 12-XX.11S: Algebraic number theory: local and p-adic fields For Galois theory see also 12-XX.11T: Finite fields and commutative rings (number-theoretic aspects)11U: Connections with logic11Y: Computational number theory, including factorization and primality testing. See also 11-0411Z05: Miscellaneous applications of number theoryThis is one of the larger fields in the Math Reviewsdatabase. Articles in Number Theory were classified with area 10 until1985, when then subfields were substantially reorganized. (It has moresub-fields than almost any other area!). From 1985-1990 there was asection 11Q: Other arithmetic-analytic topics; that material is nowusually classified with fields of analysis. There have been severalre-divisions of the overlap of this area with section 12: Field Theory(wholly subsumed in Number Theory until 1959, for example).Browse all (old) classifications for this area at the AMS.Textbooks, reference works, and tutorialsThere are many textbooks at all levels, although it is difficult tofind texts covering most of the subfields listed above; texts forthose smaller areas are of course shown on their pages.A few suggestions for texts which illustrate some of the breadth of number theory:Hardy, G. H.; Wright, E. M.: "An introduction to the theory of numbers", The Clarendon Press, Oxford University Press, New York, 1979. 426 pp. ISBN 0-19-853170-2 and 0-19-853171-0.Serre, Jean-Pierre: "A course in arithmetic", Springer-Verlag, New York-Heidelberg, 1973. 115 pp. (A humbling title!)Cohn, Harvey: "Advanced number theory", Dover Publications, Inc., New York, 1980. 276 pp. ISBN 0-486-64023-XIreland, Kenneth F.; Rosen, Michael I.: "A classical introduction to modern number theory", Springer-Verlag, New York, 1990.389 pp. ISBN 0-387-97329-XNiven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L.: "An introduction to the theory of numbers", John Wiley & Sons, Inc., New York, 1991.529 pp. ISBN 0-471-62546-9The historically inclined may prefer some basic texts of yesteryear:Landau, Edmund, "Vorlesungen über Zahlentheorie" and "Handbuch der Lehre von der Verteilung der Primzahlen", reprinted by Chelsea Publishing Co., New YorkDavenport, Harold, "The higher arithmetic -- An introduction to the theory of numbers", Dover Publications, Inc., New York, 1983. 172 pp. ISBN 0-486-24452-0Kronecker, Leopold, "Vorlesungen über Zahlentheorie", Springer-Verlag, Berlin-New York, 1978. 509 pp. ISBN 3-540-08277-8and of course we would be remiss not to suggest Gauss's Disquisitiones Arithmeticae (1801)Some distinctive resources:"Reviews in number theory, 1940-1972", six volumes (2931 pages!) edited by William J. LeVeque. American Mathematical Society, Providence, R.I., 1974"Reviews in number theory, 1973-1983", six volumes (3573 pages!) edited by Richard K. Guy. American Mathematical Society, Providence, RI, 1984. ISBN 0-8218-0218-6(A "Reviews in number theory, 1984-1996" is now also available; further information not available right now).Guy, Richard K.: "Unsolved problems in number theory", Springer-Verlag, New York, 1994. 285 pp. ISBN 0-387-94289-0 (2nd edition)Among other resources we should mention a mailing listNMBRTHRY;archives are available.Software and tablesNigel Smart has written a brief survey of the key packages for theSept. 1995 issue of the LMS Newsletter; those packages mentioned includeKANT, a system for "Computational Algebraic Number Theory"SIMATH, especially designed for number theoretic purposes.Pari, a program library and/or calculator for number theory and elliptic curves (d'après H. Cohen)Lidia, a C++ library with number-theoretic functions, your choice of underlying multiprecision kernel (freelip, gmp, libI, ...)The Magma system is well suited to number-theoretic and other algebraic structures.The general-purpose programs he mentions (Maple, Mathematica) are discussed in 68W30: Symbolic computation.A selection of software for primesSee also section 11Y: Computational number theory for issues regarding the current limits of computability in number theory (especially regarding factorizations and primality testing).Other web sites with this focusHighly recommended: the Number Theory Web. Select the most appropriate site for you: Brisbane, Australia;University of Georgia, USA;Università di Roma Tre, Rome, Italy.Algebraic Number Theory Preprint server (Illinois)Here are the AMS and Göttingen resource pages for area 11.Selected topics at this siteReview of classification categories in number theory before 1985Listing of open questions with cash rewards offered by Erdös.Yet another open Erdös problem.Other open questions worth money.Why is exp(pi*sqrt(163)) so close to an integer?The joys of the number 239Fun examples of the "law of small numbers"More examples of the "Law of Small Numbers"Even more examples of the "Law of Small Numbers"A mixed bag of number-theoretic experiments from a recent issue of Mathematics of ComputationEasy pattern to generate only prime numbers! (ha ha)A number theory question was the "hardest IMO problem, ever": When is (a^2+b^2)/(ab+1) an integer?You can reach this page through http://www.math-atlas.org/welcome.htmlLast modified 2006/07/02 by Dave Rusin. Mail: rusin@math.niu.edu |
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