1. Number Theory - Wikipedia, The Free Encyclopedia Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider http://en.wikipedia.org/wiki/Number_theory

Number theory From Wikipedia, the free encyclopedia Jump to: navigation search This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations Please improve this article by introducing more precise citations where appropriate (March 2010) In this graphic the natural numbers are arranged as in the Ulam spiral and a disk of size proportional to the number of divisors is drawn for each number: this yields an intriguing, yet not fully understood pattern. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. ( See the list of number theory topics The terms " arithmetic " or "the higher arithmetic" as nouns are also used to refer to elementary number theory. These are somewhat older terms, which are no longer as popular as they once were. However the word "arithmetic" is popularly used as an adjective rather than the more cumbersome phrase "number-theoretic", and also "arithmetic of" rather than "number theory of", e.g.

2. Number Theory - Wikibooks, Collection Of Open-content Textbooks Elementary Number Theory. This book covers an elementary introduction to Number Theory, with an emphasis on presenting and proving a large number of theorems. http://en.wikibooks.org/wiki/Number_Theory

Number Theory From Wikibooks, the open-content textbooks collection Jump to: navigation search Elementary Number Theory This book covers an elementary introduction to Number Theory, with an emphasis on presenting and proving a large number of theorems. No attempts will be made to derive number theory from set theory and no knowledge of Calculus will be assumed. It is important to convince yourself of the truth of each proof as you work through the book, and make sure you have a complete understanding. For those who wish to use this as a reference book, an index of theorems will be given. Chapters Notation and Introduction to Proof Axioms Elementary Divisibility Pythagorean Triples ... Bibliography See also Proofs in Number Theory Externals links A Computational Introduction to Number Theory and Algebra (Shoup) -retrieved Dec 21, 2009 Retrieved from " http://en.wikibooks.org/wiki/Number_Theory Subjects Number Theory Mathematics ... University level mathematics books Hidden categories: Alphabetical/N Partly developed books What do you think of this page? Please take a moment to rate this page below. Your feedback is valuable and helps us improve our website.

3. Number Theory - Encyclopedia Article - Citizendium Jul 14, 2010 Number theory is a branch of mathematics devoted primarily to the study of the integers. Any attempt to conduct such a study naturally leads http://en.citizendium.org/wiki/Number_theory

Number theory From Citizendium, the Citizens' Compendium Jump to: navigation search addthis_pub = 'citizendium'; addthis_logo = ''; addthis_logo_color = ''; addthis_logo_background = ''; addthis_brand = 'Citizendium'; addthis_options = ''; addthis_offset_top = ''; addthis_offset_left = ''; Main Article Talk Related Articles Bibliography External Links This is a draft article , under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to edit intro Contents Origins The dawn of arithmetic Classical Greece and the early Hellenistic period Diophantus ... External links Number theory is a branch of mathematics devoted primarily to the study of the integers. Any attempt to conduct such a study naturally leads to an examination of the properties of that which integers are made out of (namely, prime numbers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalisations of the integers (such as, for example, algebraic integers). Integers can be considered either as such or as solutions to equations ( diophantine geometry ). Some of the main questions are those of distribution: questions, say, on patterns or their absence (in the primes or other sequences) or, more generally, questions on size, number and growth. Such matters are often best understood through the study of

4. Number Theory - Simple English Wikipedia, The Free Encyclopedia Number theory is a part of mathematics. It explains what some types of http://ja.wikipedia.org/wiki/simple:Number_theory

Number theory From Wikipedia, the free encyclopedia Jump to: navigation search Number theory is a part of mathematics . It explains what some types of numbers are, what properties they have, and ways that they can be useful. Topics in number theory are: Prime numbers Factorization Important theorems in number theory are: Fundamental theorem of arithmetic Fermat's last theorem Fermat's little theorem change Applications A well-known application of number theory is encrypted messaging ( encryption Data compression also makes use of the field. This short article about mathematics or a similar topic can be made longer. You can help Wikipedia by adding to it Retrieved from " http://simple.wikipedia.org/wiki/Number_theory Category Number theory Hidden category: Math stubs Personal tools New features Log in / create account Namespaces Page Talk Variants Views Read Change View history Actions Search Getting around Main Page Simple start Simple talk New changes ... Give to Wikipedia Print/export Make a book Download as PDF Page for printing Toolbox What links here Related changes Special pages Permanent link ... Cite this page In other languages Aragonés Asturianu Brezhoneg Català ... Võro This page was last changed on 14 October 2010, at 17:23.

5. Category:Number Theory - Wikipedia, The Free Encyclopedia Traditionally, number theory is the branch of mathematics concerned with the http://en.wikipedia.org/wiki/Category:Number_theory

Category:Number theory From Wikipedia, the free encyclopedia Jump to: navigation search Wikibooks has a book on the topic of Number Theory The main article for this category is Number theory Traditionally, number theory is the branch of mathematics concerned with the properties of integers and many of its open problems are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arise naturally from the study of integers. Number theory may be subdivided into several fields according to the methods used and the questions investigated. See for example the list of number theory topics This category corresponds roughly to MSC 11 Number Theory in the American Mathematical Society 's Mathematics Subject Classification Wikimedia Commons has media related to: Number theory (previous 200) ( next 200 Subcategories This category has the following 19 subcategories, out of 20 total. A Algebraic number theory (4 C, 105 P) Analytic number theory (5 C, 72 P) Arithmetic functions (2 C, 18 P)

6. Partition (number Theory) - Wikipedia, The Free Encyclopedia In number theory, a partition of a positive integer n is a way of writing n http://en.wikipedia.org/wiki/Partition_(number_theory)

Partition (number theory) From Wikipedia, the free encyclopedia Jump to: navigation search Ferrers diagrams showing the partitions of positive integers 1 through 8. They are so arranged that images under the reflection about the main diagonal of the square are conjugate partitions. In number theory , a partition of a positive integer n is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a composition . A summand in a partition is also called a part . The number of partitions of n is given by the partition function p n Contents Examples Partition function Intermediate function Generating function ... edit Examples The partitions of 4 are listed below: The partitions of 8 are listed below: edit Partition function In number theory , the partition function p n ) represents the number of possible partitions of a natural number n , which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. For example, 4 can be partitioned in five distinct ways:

7. The Math Forum - Math Library - Number Theory The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/number_theory/

Browse and Search the Library Home Math Topics : Number Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Algebraic Number Theory Analytic Number Theory Sequences and Sets Fibonacci Sequence Selected Sites (see also All Sites in this category Continued Fractions: an Introduction - Adam Van Tuyl A brief introduction to the field of continued fractions, including some basic theory about the subject; the history of continued fractions, tracing some of the major developments in the field in the past 2500 years; some interactive applications that demonstrate the uses of continued fractions and let you calculate them; and the resources used in creating this site, including a bibliography and links to other sites on the Web. more>> Fermat's Last Theorem - MacTutor Math History Archives Essay describing Fermat's theorem with links to mathematicians such as Sophie Germain, Legendre, Dirichlet, Shimura and Taniyama, etc., from its inception through Andrew Wiles' proof, with another web site and 17 references (books/articles). more>> Number Theory - Dave Rusin; The Mathematical Atlas

8. UM Mathematics-Number Theory Graduate Program in Number Theory. Staff, research interests, courses, seminars. http://www.math.lsa.umich.edu/research/number_theory/

Number Theory at the University of Michigan Faculty: Jeff Lagarias Algebraic and analytic number theory, Diophantine approximation. Hugh Montgomery Analytic number theory, distribution of prime numbers, Fourier analysis, analytic inequalities, probability. Kartik Prasanna Arithmetic of automorphic forms and periods, special values of L-functions, Iwasawa theory. Michael Zieve Algebraic number theory and arithmetic geometry, especially Galois theory and arithmetic dynamics. Faculty in related areas: Mitya Boyarchenko and Stephen DeBacker (representation theory); Igor Dolgachev William Fulton Robert Lazarsfeld , and Karen Smith (algebraic geometry); Harm Derksen and Gopal Prasad (algebraic groups). Postdocs: Michael Khoury Ruochuan Liu Chris Lyons Graduate students (advisor): Hunter Brooks (Prasanna), Sijun Liu (Zieve), Andrey Mishchenko (Lagarias), Alex Mueller (Zieve), Hieu Ngo (Montgomery), Julian Rosen (Lagarias), Zach Scherr (Zieve)

9. Number Theory - Definition Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily http://www.wordiq.com/definition/Number_theory

Number theory - Definition Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers. Number theory may be subdivided into several fields according to the methods used and the questions investigated. See for example the list of number theory topics Mathematicians working in the field of number theory are called number theorists The term " arithmetic " is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Nevertheless, the term remains prevalent e.g. in the names of mathematical fields (arithmetic algebraic geometry and the arithmetic of elliptic curves and surfaces). This sense of the term arithmetic should not be confused with the branch of logic which studies arithmetic in the sense of formal systems. In elementary number theory , the integers are studied without use of techniques from other mathematical fields. Questions of

10. Analytic Number Theory - Wikipedia, The Free Encyclopedia In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have http://en.wikipedia.org/wiki/Analytic_number_theory

Analytic number theory From Wikipedia, the free encyclopedia Jump to: navigation search This article may require cleanup to meet Wikipedia's quality standards Please improve this article if you can. The talk page may contain suggestions. (June 2008) This article needs additional citations for verification Please help improve this article by adding reliable references . Unsourced material may be challenged and removed (September 2008) In mathematics analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet 's introduction of Dirichlet L -functions to give the first proof of Dirichlet's theorem on arithmetic progressions Another major milestone in the subject is the prime number theorem Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. Multiplicative number theory deals with the distribution of the prime numbers , such as estimating the number of primes in an interval, and includes theorems such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

11. Number Theory | TripAtlas.com Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that http://tripatlas.com/Number_theory

12. The Math Forum - Math Library - Number Theory The Math Forum's Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites relating to Number http://www.mathforum.org/library/topics/number_theory/

Browse and Search the Library Home Math Topics : Number Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Algebraic Number Theory Analytic Number Theory Sequences and Sets Fibonacci Sequence Selected Sites (see also All Sites in this category Continued Fractions: an Introduction - Adam Van Tuyl A brief introduction to the field of continued fractions, including some basic theory about the subject; the history of continued fractions, tracing some of the major developments in the field in the past 2500 years; some interactive applications that demonstrate the uses of continued fractions and let you calculate them; and the resources used in creating this site, including a bibliography and links to other sites on the Web. more>> Fermat's Last Theorem - MacTutor Math History Archives Essay describing Fermat's theorem with links to mathematicians such as Sophie Germain, Legendre, Dirichlet, Shimura and Taniyama, etc., from its inception through Andrew Wiles' proof, with another web site and 17 references (books/articles). more>> Number Theory - Dave Rusin; The Mathematical Atlas

13. Elementary Number Theory — Sage Constructions V4.5.3 Table Of Contents. Elementary number theory. Taking modular powers; Discrete logs; Prime numbers; Divisors; Quadratic residues http://www.sagemath.org/doc/constructions/number_theory.html

Navigation index next previous Sage Constructions v4.5.3 Elementary number theory Taking modular powers How do I compute modular powers in Sage? To compute in Sage, type sage: R Integers sage: a R sage: a Instead of R Integers(97) you can also type R IntegerModRing(97) . Another option is to use the interface with GMP: sage: powermod Discrete logs To find a number such that (the discrete log of log command: sage: r Integers sage: b r sage: a b sage: a log b This also works over finite fields: sage: FF FiniteField "a" sage: a FF gen sage: c a sage: c log a Prime numbers How do you construct prime numbers in Sage? The class Primes allows for primality testing: sage: in Primes False sage: in Primes True The usage of is self-explanatory: sage: The Pari command primepi is used via the command . This returns the number of primes , for example: sage: pari primepi Using or primes one can check that, indeed, there are primes up to sage: sage: list primes Divisors How do you compute the sum of the divisors of an integer in Sage? Sage uses divisors(n) for the number (usually denoted ) of divisors of and sigma(n,k)

14. Number Theory This site area outlines a development of the properties of real and complex numbers from counting (enumeration) and geometric assumptions. http://whyslopes.com/Number_Theory/

Appetizers and Lessons for Ma thematics and Reason hire Help Haiti. Help flood victims in Pakistan - steps to develop skills - adults welcome Logique Definition d'une variable Parents- Help your child or teen with Speaking skills Preparing for science Ends, math skill building booklets ... for 4 to 13 (parent friendly) Calculus - appetizers (i) why study slopes and (ii) why factor polynomials Calculus: learn more Online-Books order Elements of Reason. Pattern Based Reason for avid readers Math. Curriculum Notes what needed to be address Three Skills for Algebra words and stories to improve logic and algebra, and preparation for calculus - steps to ease or avoid common difficulties in calculus. Key, Key, Key Lessons 2 logic puzzles mathematical. induction painless theorem proving Pythagorean theorem ... show work formats (vertical alignment of = signs) to improve performance K5 to college level. High School Mathematics Steps A to J a solid base if only you digest it.

15. Number Theory - Simple English Wikipedia, The Free Encyclopedia Number theory is a part of mathematics. It explains what some types of numbers are, what properties they have, and ways that they can be useful. Topics in number theory are http://simple.wikipedia.org/wiki/Number_theory

Number theory From Wikipedia, the free encyclopedia Jump to: navigation search Number theory is a part of mathematics . It explains what some types of numbers are, what properties they have, and ways that they can be useful. Topics in number theory are: Prime numbers Factorization Important theorems in number theory are: Fundamental theorem of arithmetic Fermat's last theorem Fermat's little theorem change Applications A well-known application of number theory is encrypted messaging ( encryption Data compression also makes use of the field. This short article about mathematics or a similar topic can be made longer. You can help Wikipedia by adding to it Retrieved from " http://simple.wikipedia.org/wiki/Number_theory Category Number theory Hidden category: Math stubs Personal tools New features Log in / create account Namespaces Page Talk Variants Views Read Change View history Actions Search Getting around Main Page Simple start Simple talk New changes ... Give to Wikipedia Print/export Make a book Download as PDF Page for printing Toolbox What links here Related changes Special pages Permanent link ... Cite this page In other languages Aragonés Asturianu Brezhoneg Català ... Võro This page was last changed on 14 October 2010, at 17:23.

16. Number Theory - AoPSWiki Dec 30, 2009 Number theory is the field of mathematics associated with studying the properties and identities of real numbers. http://www.artofproblemsolving.com/Wiki/index.php/Number_theory

Art of Problem Solving LOGIN/REGISTER Home Bookstore AoPS Curriculum ... Articles You are here: Resources AoPSWiki Search Wiki Page Tools Page Discussion View source History Personal tools Talk for this IP Log in Navigation Main Page Community portal Current events Recent changes ... Help Toolbox What links here Related changes Special pages Printable version ... Permanent link Number theory From AoPSWiki Number theory is the field of mathematics associated with studying the properties and identities of real numbers Contents Overview Student Guides to Number Theory Resources Books ... Famous Unsolved Number Theory Problems Overview Number theory is a broad topic, and may cover many diverse subtopics, such as: Modular arithmetic Prime numbers Some branches of number theory may only deal with a certain subset of the real numbers, such as integers positive numbers, natural numbers rational numbers , etc. Some algebraic topics such as Diophantine equations are occasionally considered number theory. Student Guides to Number Theory Introductory topics in number theory Covers different kinds of integers such as prime numbers composite numbers , and their relationships ( multiples divisors , and more). Also includes

17. Number Theory At The University Of Georgia Number Theory and Arithmetic Geometry Group. Members, seminars. http://www.math.uga.edu/~lorenz/Number_Theory_Group.html

Number Theory and Arithmetic Geometry Group Permanent faculty and their fields of interests. William Alford Associate Professor, Ph.D. Tulane, 1963. Factoring and other number theory problems by computer. Matthew Baker Assistant Professor, Ph.D. U.C. Berkeley, 1999. Galois actions on torsion points. Modular curves and their Jacobians. Discreteness problems for arithmetic heights. Linear series and vector bundles in characteristic p. Arithmetic of curves and their Jacobians. Sybilla Beckmann Associate Professor, Ph.D. U. Penn., 1986. Galois theory. The inverse galois problem, that is, to determine whether every finite group is the galois group of some extension of the rational numbers. Arithmetic information on branched coverings, such as fields of definition. Tilings of the plane. Mathematics education. Andrew Granville Barrow Professor, Ph.D. Queens,1987. Distribution of primes. Sieving intervals. Distribution of `smooth' numbers. Properties of binomial coefficients. Cyclotomic fields. Carmichael numbers. Exponential sums. Integer solutions to Diophantine equations. Binary quadratic forms and the elementary theory of elliptic curves. Questions related to factoring and primality testing. Symbolic computation and `computing by homomorphisms'. Computational complexity, particularly lower bounds. Power series and the combinatorics of coefficients. Counting lattice points. Dan Lieman Associate Professor, Ph.D. Brown, 1992.

18. Long Division Method After The Decimal Point Long Division method provides a sequence of decimal approximations to a fraction M/N. If the sequence is finite the fraction is decimal. http://whyslopes.com/Number_Theory/Decimals_Long_Division.html

Appetizers and Lessons for Ma thematics and Reason hire Help Haiti. Help flood victims in Pakistan - steps to develop skills - adults welcome Logique Definition d'une variable Parents- Help your child or teen with Speaking skills Preparing for science Ends, math skill building booklets ... for 4 to 13 (parent friendly) Calculus - appetizers (i) why study slopes and (ii) why factor polynomials Calculus: learn more Online-Books order Elements of Reason. Pattern Based Reason for avid readers Math. Curriculum Notes what needed to be address Three Skills for Algebra words and stories to improve logic and algebra, and preparation for calculus - steps to ease or avoid common difficulties in calculus. Key, Key, Key Lessons 2 logic puzzles mathematical. induction painless theorem proving Pythagorean theorem ... show work formats (vertical alignment of = signs) to improve performance K5 to college level. High School Mathematics Steps A to J a solid base if only you digest it.

19. Counting Whole No. Factors Number Theory Practices Counting Whole No. Factors. A. Start of Number http://whyslopes.com/Number_Theory/Prime_Decomposition_Generating_And_Counting_F

Appetizers and Lessons for Ma thematics and Reason hire Help Haiti. Help flood victims in Pakistan - steps to develop skills - adults welcome Logique Definition d'une variable Parents- Help your child or teen with Speaking skills Preparing for science Ends, math skill building booklets ... for 4 to 13 (parent friendly) Calculus - appetizers (i) why study slopes and (ii) why factor polynomials Calculus: learn more Online-Books order Elements of Reason. Pattern Based Reason for avid readers Math. Curriculum Notes what needed to be address Three Skills for Algebra words and stories to improve logic and algebra, and preparation for calculus - steps to ease or avoid common difficulties in calculus. Key, Key, Key Lessons 2 logic puzzles mathematical. induction painless theorem proving Pythagorean theorem ... show work formats (vertical alignment of = signs) to improve performance K5 to college level. High School Mathematics Steps A to J a solid base if only you digest it.

20. Number Theory: Facts, Discussion Forum, And Encyclopedia Article Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty http://www.absoluteastronomy.com/topics/Number_theory

Home Discussion Topics Dictionary ... Login Number theory Number theory Topic Home Discussion Definition Discussion Ask a question about ' Number theory Start a new discussion about ' Number theory Answer questions from other users Full Discussion Forum Encyclopedia Number theory is the branch of pure mathematics Pure mathematics Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty... concerned with the properties of number Number A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number... s in general, and integer Integer The integers are formed by the natural numbers including together with the negatives of the non-zero natural numbers... s in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (

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