41. Algorithmic Number Theory Tables And Links Compiled by Noam Elkies.Category Science Math number theory Tables......Algorithmic (aka Computational) number theory Tables, Links, etc. Algorithmicnumber theory? What's that? Back to general math page http://www.math.harvard.edu/~elkies/compnt.html

42. Number Theory Foundation Private philanthropic US organization for research in number theory. http://www.cerias.purdue.edu/homes/ssw/ntf/index.html

Number Theory Foundation An Announcement about the Number Theory Foundation. How to apply for a grant from the Number Theory Foundation. Send e-mail to Sam Wagstaff (This page last modified January 7, 1999)

43. Basics Of Computational Number Theory Notes and Javascript illustrations by Robert Campbell.Category Science Math number theory Computational......Basics of Computational number theory. Robert Campbell. Contents. 1. Introduction.This document is a gentle introduction to computational number theory. http://www.math.umbc.edu/~campbell/NumbThy/Class/BasicNumbThy.html

Basics of Computational Number Theory Robert Campbell Contents Introduction Modular Arithmetic GCD - The Euclidean Algorithm Exponentiation - The Russian Peasant Algorithm ... Galois Fields Appendices Programming Notes References Glossary Introduction This document is a gentle introduction to computational number theory. The plan of the paper is to first give a quick overview of arithmetic in the modular integers. Throughout, we will emphasize computation and practical results rather than delving into the why. Simple programs, generally in JavaScript, are available for all of the algorithms mentioned. At the end of the paper we will introduce a the Gaussian Integers and Galois Fields and compare them to the modular integers. Companion papers will examine number theory from a more advanced perspective. Modular Arithmetic Modular arithmetic is arithmetic using integers modulo some fixed integer N . Thus, some examples of operations modulo 12 are: 7 + 7 = 14 = 2 (mod 12) 5 * 7 = 35 = 11 (mod 12) Further examples can be generated and checked out with the following short programs. Note that, as JavaScript cannot compute with integers larger than 20 digits, the largest modulus allowed is 10 digits. (mod (mod Among the basic operations we have missed the division operator. If we were working in the integers we would almost never be able to define a quotient (unless the answer is itself an integer). In the modular integers we can often, but not always, define a quotient:

44. Inexplicable Secrets Of Creation Relationships between number theory and physics. http://www.maths.ex.ac.uk/~mwatkins/

"Upon looking at these numbers, one has the feeling of being in the presence of one of the inexplicable secrets of creation [D. Zagier] mysterious occurrences on the interface of physics and number theory introductory prime number theory resources related curiosities weblog ... a message about the future of this website

45. Algorithmic Number Theory Eric Bach and Jeffrey Shallit. Errata, bibliography in BibTeX format.Category Science Math number theory Publications Books......Algorithmic number theory. Eric Bach and Jeffrey Shallit Algorithmicnumber theory, Volume I Efficient Algorithms Published by MIT http://www.math.uwaterloo.ca/~shallit/ant.html

Algorithmic Number Theory Eric Bach and Jeffrey Shallit Algorithmic Number Theory, Volume I: Efficient Algorithms Published by MIT Press , August 1996 xvi + 512 pages US $55.00 ISBN 0-262-02405-5 (v.1) Library of Congress Call Number QA 241.B1085 1996 Description and Table of Contents What Some Readers and Reviewers Have Said Bibliography Errata and Addenda ... When will Volume 2 be published? E-mail:

46. Algorithmic Number Theory - Bibliography BibTeX files for the book Algorithmic number theory by Eric Bach and Jeffrey Shallit.Category Science Math number theory Publications......Algorithmic number theory Bibliography. We hope this database of over 3100 papersrelating to algorithmic number theory will be useful to other researchers. http://www.math.uwaterloo.ca/~shallit/antbib.html

Algorithmic Number Theory - Bibliography This page gives you access to the BibTeX files for the book Algorithmic Number Theory by Eric Bach and Jeffrey Shallit. We hope this database of over 3100 papers relating to algorithmic number theory will be useful to other researchers. We have double-checked most of these references, but we cannot guarantee their accuracy. Please inform us of errors or omissions you find. Bib files corresponding to chapters of our book not yet written (such as factoring.bib ) are necessarily spotty and incomplete. The bib files do not contain duplicate entries, so finding a particular reference may require checking several different files, particularly when the subject matter spans two different sub-areas. Please note that the BibTeX references use abbreviations for names of journals, conferences, and publishers, and these abbreviations are stored in the file abbrevs.bib abbrevs.bib (size = 21090 bytes) algebraic.bib (size = 10987 bytes) analytic.bib (size = 113330 bytes) applications.bib

47. Number Theory Department of number theory. Members, research interests, publications. http://www.impan.gov.pl/FacInfo/numberth.html

Number Theory Staff Research during the last few years concerned problems in different branches of number theory which will be reviewed in the order adopted by Mathematical Reviews. In elementary number theory a problem (proposed by T. Cochrane and G. Meyerson) concerning covering systems of congruences has been solved in [10] and another one (proposed by W. Narkiewicz) concerning arithmetical functions in [14]. Various problems concerning generalized pseudoprimes have been solved in [1]-[4] and a problem on this subject proposed by C. Pomerance has been solved by A. Rotkiewicz (as yet unpublished). A criterion for reducibility over the rationals of the non-cyclotomic kernel of a non-reciprocal lacunary polynomial has been given by A. Schinzel (as yet unpublished). In diophantine equations, [8] gives a solution to Problem D16 from the book of R. Guy "Unsolved Problems in Number Theory" and [15] solves a problem on Pythagorean triangles proposed by I. Korec. [16] deals with a Diophantine equation related to generalized Bernoulli numbers. In the metric theory of algorithms a problem of M. Deleglise concerning continued fractions has been solved in [13].

48. Fields Institute - Conference In Number Theory - 2003 Conference in number theory in Honour of Professor HC Williams. SaturdayMay 24, 2003 to Friday May 30, 2003 to be held at The Banff http://www.fields.utoronto.ca/programs/scientific/02-03/numtheory/

SCIENTIFIC ACTIVITIES March 18, 2003 Home About Us Mathematics Education Calendar of Events ... Search Conference in Number Theory in Honour of Professor H.C. Williams Saturday May 24, 2003 to Friday May 30, 2003 to be held at The Banff Centre , Banff, Alberta, Canada Sponsored by The Fields Institute, The Number Theory Foundation RSA Security Inc. iCore CISaC , and The University of Calgary Registration Confirmed Attendees Manindra Agrawal Lecture ... Conference Proceedings Organizing Committee Michael Jacobson, Calgary Renate Scheidler, Calgary Jon Sorenson, Butler Andreas Stein, UIUC Gary Walsh, Ottawa Summary The conference is open to all areas of Number Theory, with emphasis on Computational Number Theory and applications to Cryptography. Researchers in these fields of study are welcomed to participate, as we honour Canada's foremost computational number theorist, whose contributions include results on integer factorization, primality testing, diophantine equations, linear recurrences, the infrastructure of quadratic number fields and function fields, and their applications to Cryptography. Each day of the conference will include one plenary lecture, a number of invited lectures and time has been set aside each day for contributed talks. Special Lecture Manindra Agrawal from IITK, who has recently proved the existence of a deterministic polynomial time algorithm for primality testing, will give a special evening lecture at the Banff Center on Sunday May 25, 2003, with a reception to follow in his honour sponsored by RSA Security Inc.

49. Math Resources Contains links to software, professional organizations, research institutes, teaching aids, algebra, number theory, geometry, calculus, differential equations. http://comers.citadel.edu/www/math.htm

Math Resourses This page contains links to web sites with mathematically related information organized as follows: General Resource sites Professional Organizations and Research Institutes Math Teaching Resources Algebra ( including Lattice Theory), Discrete Math and Number Theory ... Other General Resource sites The MathSci Net (On-campus access) The list of Mathematics Information Servers maintained by Penn State University The Mathematics Archives WWW Server at the University of Tennessee The World-Wide Web Virtual Library: Mathematics Mathematical Resources on the Web , from Univ Florida E-Tutor Connected Learning for K-12 Centre for the Popularisation of Mathematics Interactive Math , Miscellany and puzzles Spreadsheets in Education and Mathematics (Erich Neuwirth) Spreadsheet User , a journal devoted to novel spreadsheet applications Mega Math from Los Alamos Nat. Lab MatLab in Education and MatLab Connections listing third-party tools MathSkills Newsletters from the British Dept of Education and Employment and MathSkills home page Mathematical PREprint Server System by the European Math Soc Math Placement and Gateway Exams from Michagan State University Organics Math Workshop Conference proceedings Home pages for professional organizations and research institutes: AMS , the MAA the Eisenhower National Clearinghouse and EXTEND an Internet forum on math education.

50. The Math Forum - Math Library - Number Theory mathematics. This page contains sites relating to number theory. Browseand Search the Library Home Math Topics number theory. 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

51. Tools On Number Theory Web æ•´æ•°è«–é–¢é€£ã®ã‚³ãƒ³ãƒ”ãƒ¥ãƒ¼ã‚¿ãƒ„ãƒ¼ãƒ«ã®ã‚¢ãƒ¼ã‚«ã‚¤ãƒ–ã¨ãƒªãƒ³ã‚¯é›†ã€‚ã€Œä»£æ•°å­¦ã¨è¨ˆç®—ã€ç ”ç©¶é›†ä¼šã®æƒ å ±ã‚‚ã‚ã‚‹ã€‚ http://tnt.math.metro-u.ac.jp/index.j.html

English Tools on Number Theory Web <%8(B(Unofficial!) <%899?7MzNr(B TNT$B$K$D$$$F(B TNT Mailing List archive TNT FTP $B%3%s%F%s%D>R2p(B <0%_%i! <%5%$%H0lMw(B TNT Web $B4XO"%j%s%/(B $B$3$N%Z! ogino@tnt.math.metro-u.ac.jp )$B$K$h$C$F%a%s%F%J%s%9$5$l$F$$$^$9!#(B $B$40U8+$4MWK>$r$*4s$;2 <$5$$!#(B <0%Z! <%8(B $B$X$N$40U8+$O!"(B webmaster@tnt.math.metro-u.ac.jp $B$^$G$*4j$$CW$7$^$9!#(B $BI=;f(B $B99?7(B $B%j%s%/(B $BG[I[J*(B ] [TNT$B$K$D$$$F(B] ogino@tnt.math.metro-u.ac.jp $Id: index.j.html,v 1.1.1.1 2003/01/10 02:07:28 root Exp $

52. Math Forum - Ask Dr. Math Archives: High School Number Theory Browse High School number theory. Stars indicate particularly interestinganswers or good places to begin browsing. 0 Divided by http://mathforum.org/library/drmath/sets/high_number_theory.html

Ask Dr. Math High School Archive Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ TOPICS This page: number theory Search Dr. Math See also the Dr. Math FAQ 0 to power n to power dividing by 0 number bases Internet Library number theory HIGH SCHOOL About Math Analysis Algebra basic algebra ... Trigonometry Browse High School Number Theory Stars indicate particularly interesting answers or good places to begin browsing. 0 Divided by 0 What is the answer to divided by 0? I think it is undefined... 100 Factorial in Base 6: How Many Zeros? How many zeros are there at the end of 100! in base 6? Adding Arithmetic Sequences How do you add the numbers from 1 to 5000 without actually doing it or using a calculator? What if you were adding just the odd numbers? Adding Hexadecimals A complete introduction to the hexadecimal, including sample problems on addition. Adding in Base 9 and Base 5 In 3rd grade my son is learning how to add in bases other than 10. Arithmetic in Other Bases I need to know how to write out the steps for addition, subtraction, multiplication, and division in other bases. Base 12 Fractions I need a simple way to understand how to do and interpret decimals in base 12.

53. Paul Garrett: Crypto And Number Theory Introduction to cryptology, number-theory, and algorithms. Protocols. Symmetric versus asymmetric Category Science Math number theory Publications Books......Crypto and number theory. garrett@math.umn.edu my homepage Introductionto cryptology, numbertheory, algebra, and algorithms. Protocols. http://www.math.umn.edu/~garrett/crypto/

Crypto and Number Theory garrett@math.umn.edu my homepage updated 19 Dec 02] [this page is http://www.math.umn.edu/~garrett/crypto/] Office hours in finals week: Mon Dec 16 3:00-4:00 Tue Dec 17 4:00-5:00 Wed Dec 18 3:00-4:00 The final is Thurs Dec 19 1:30-3:30 in our usual room Grades as they're available Overheads Primality vs. factoring, primality certificates: Wed Nov 06, 2002 Rijndael: Wed Nov 27, 2002 Quiz solutions: s01.pdf s02.pdf s03.pdf s04.pdf ... s11.pdf This course uses my book I've developed for this course. ... [ some errata in first printing remaining errata in second printing Snippets of code for basic number-theoretic algorithms . A little interactive stuff , including some small computational assists in lieu of other computing resources. Introduction to cryptology, number-theory, algebra, and algorithms. Protocols. Symmetric versus asymmetric systems. Stream, block ciphers. One-way functions, signatures. Key management issues. DES, AES (Rijndael). (Pseudo-) random number generation. Permutation groups, primes, Euclidean algorithm, finite fields, quadratic reciprocity. Discrete logs, RSA, pseudoprimes, rho method. Elliptic curve methods. Quadratic sieve. And so on and so on... [3:35-5:00 in Science Classroom Building 175] Supplementary references/sources/links Grading: We'll have a take-home quiz each week (given out Monday, collected Wednesday)

54. The USC Number Theory Home Page s in number theory Description number theory Group. Members, research interests, courses, links.Category Science Math number theory Research Groups...... I also have interests in the application of these results to gap problemsin number theory. Graduate Course http://www.math.sc.edu/~filaseta/numthry.html

Number Theorists On Our Faculty Graduate Course Descriptions (in Number Theory) More About Our Department And University Some Other Links Of Interest ... Michael Filaseta Research Interests: Number Theory, including Analytic, Classical Algebraic, Combinatorial, Computational, Elementary, and Transcedence topics. I have particular interests in results associated with lattice points close to (or on) a curve or surface, the distribution of special sequences of integers in short intervals, applications of Pade approximations to Number Theory, the irreducibility of polynomials over the rationals, and computations with sparse or lacunary polynomials. Richard Hudson Research Interests: elementary number theory, analytic prime number theory, quadratic forms, class number formulae, forms of higher order, quadratic and higher power residues, comparative prime number theory, Gauss and Jacobi sums, computer results in number theory. Kate Hurley Research Interests: My research concerns modularity properties of VOAs. One can associate a q-graded function called a graded trace or one-point correlation function to a given VOA, and in some cases these functions are modular forms. I have worked on determining which modular forms are realized as graded traces. An additional recent interest of mine is inverse Galois theory. Charles Nicol (Retired but still collaborating in research) Research Interests: aspects of Computational and Elementary Number Theory, including work on cyclotomic polynomials, Euler's phi function, and problems related to digits in integer sequences.

55. Math 780 Material Lecture notes by Michael Filaseta, University of South Carolina, Fall 1997 (PS).Category Science Math number theory Education......Math 780 Elementary number theory. A postcript version of the coursedescription can be obtained below. Class notes (55 pages) can http://www.math.sc.edu/~filaseta/gradcourses/Math780.html

Math 780: Elementary Number Theory A postcript version of the course description can be obtained below. Class notes (55 pages) can also be obtained in postscript form below. These notes are from a course taught by Michael Filaseta in the Fall of 1997 and may not reflect the current semesters material. A graph, at the bottom of the page, compares Li(x), Pi(x), and x/log(x). Math 780 Course Description (postscript file) Math 780 Class Notes A Page For Our Undergraduate Elementary Number Theory Course Graduate Number Theory Courses At The University of South Carolina

56. Course 311 - Abstract Algebra Lecture notes by David Wilkins, Trinity College, Dublin. Topics in number theory; Group Theory; Galois Theory. http://www.maths.tcd.ie/~dwilkins/Courses/311/

Course 311 - Abstract Algebra The lecture notes for course 311 ( Abstract algebra ), taught at Trinity College, Dublin, in the academic year 2001-02, are available here. The course consists of three parts:- Part I: Topics in Number Theory DVI PDF PostScript Part II: Topics in Group Theory DVI PDF PostScript Part III: Introduction to Galois Theory DVI PDF PostScript The following handouts were also distributed in the academic year 2001-02: A collection of problems DVI PDF PostScript The resolvent cubic of a quartic polynomial DVI PDF PostScript dwilkins@maths.tcd.ie ... Trinity College , Dublin 2, Ireland dwilkins@maths.tcd.ie

57. Algebraic Number Theory Course notes by Robin Chapman, University of Exeter, May 2000.Category Science Math number theory Algebraic......MAS4002 Algebraic number theory. This is the home page for the Algebraicnumber theory course. At present it is still under construction. http://www.maths.ex.ac.uk/~rjc/courses/ant99/ant99.html

MAS4002: Algebraic Number Theory This is the home page for the Algebraic Number Theory course. At present it is still under construction. Eventually it will contain copies of course handouts, commentaries on some of my more challenging problems, and useful links. Most files are in dvi format. The course will be closely based on the following book: Ian Stewart and David Tall, Algebraic Number Theory , Chapman and Hall. Alas the paperback is out of print (and the hardback is overpriced) but the following alternative is very cheap (but a bit old-fashioned): Harry Pollard and Harold G. Diamond, The Theory of Algebraic Numbers , Dover. Course information Background on rings and ideals 1998 exam Problem sheet 1 ... Almost complete summary of notes to course (version 5, recently corrected!) Robin Chapman Room 811, Laver Building University of Exeter Exeter, EX4 4QE, UK rjc@maths.ex.ac.uk 2nd May 2000 Back to teaching page Back to home page

58. Jim Loy's Mathematics Page Articles on geometry, algebra, number theory, and many other branches of mathematics. http://www.jimloy.com/math/math.htm

Go to my home page Jim Loy's Mathematics Page "He must be a 'practical' man who can see no poetry in mathematics." - W.F. White Dedicated to the memory of Isaac Asimov. See the top of my Science pages for comments on Dr. Asimov. My Mathematics pages were described briefly in the Math Forum Internet News No. 5.48 (27 November 2000) There are no uninteresting numbers. Assume that there are. Then there is a lowest uninteresting number. That would make that number very interesting. Which is a contradiction. Contents: Algebra Geometry Calculus and Pre-Calculus Arithmetic, Roman Numerals ... ... Other Links Algebra: Intro: Before You Took Algebra (includes the most basic laws of algebra) The Essence of Algebra (equations and variables) Word Problems Distributive Law Intro to Matrices Determinants ... Square Both Sides? or (x+y)^2 Quadratic Formula Cubic Equations Irrational Numbers e (also under Calculus and Pre-Calculus) Logarithms (also under Calculus and Pre-Calculus) The Slide Rule The Words of Mathematics (basic glossary) Sequences and Series: Infinite Series (Identify your series here) Geometric Series Fibonacci Numbers Harmonic Series Two Infinite Series ... A Leaning Tower (also under Geometry) Zeno's Paradoxes in my Physics pages Complex (and Imaginary) Numbers: Imaginary Numbers (and complex numbers) Roots of Unity Gaussian Primes Others: Surprise?

59. Number Theory Lecture notes by Nigel Byott, University of Exeter (DVI,PDF,PS).Category Science Math number theory Education......Module Homepage for MAS3008 number theory 2002/2003. Old examination papers.number theory Examination, Summer 2002, as .dvi or .ps or .pdf. http://www.maths.ex.ac.uk/~NPByott/teaching/NT.html

Module Homepage for MAS3008 Number Theory 2002/2003 A handout giving details of lecture times, assessment schedule, brief syllabus, booklist etc, is available here as .dvi or .ps or .pdf The Examples Sheets and other handouts will appear here (as .dvi, .ps or .pdf files) as the module progresses. Examples Sheets: Examples Sheet 1, as .dvi or .ps or .pdf Examples Sheet 2, as .dvi or .ps or .pdf Examples Sheet 3, as .dvi or .ps or .pdf Examples Sheet 4, as .dvi or .ps or .pdf Examples Sheet 5, as .dvi or .ps or .pdf Solutions to Examples Sheets: Solutions to Examples Sheet 1, as .dvi or .ps or .pdf Solutions to Examples Sheet 2, as .dvi or .ps or .pdf Solutions to Examples Sheet 3, as .dvi or .ps or .pdf Solutions to Examples Sheet 4, as .dvi or .ps or .pdf Solutions to Examples Sheet 5, as .dvi or .ps or .pdf MAPLE work. Optional MAPLE Exercises, as .dvi or .ps or .pdf Executable MAPLE Worksheet with solutions to these exercises. You will probably need to save this as a file in order to run it in MAPLE. Same thing, as

60. Mathematical Induction - Math Induction Problems And Puzzles A page of uncommon problems, most closely connected with number theory. http://www.geocities.com/jespinos57/induction.htm

MATHEMATICAL INDUCTION MATHEMATICAL INDUCTION Math Links PDF format Naoki Sato's Solutions PDF Spanish Version connected with Number Theory.Some properties may be proved in different ways.For more exercises, problems, puzzles, games, math riddles, brain teasers, etc. to see Math Links (not only of math induction). PROBLEM 1 Let : F(n)= p-1 k=1 k n(p-1)+1 -n(n-1) p-1 k=1 ((k )/2) - p(p-1)(n(p-1)+1)/2 Prove by induction that F(n) is divisible by p , for all integers n Hint PROBLEM 2 Use mathematical induction to prove the following: FIB(n) (FIB(n+2)=FIB(n+1)+FIB(n); FIB(1)=1,FIB(2)=1. Fibonacci sequence) Hint PROBLEM 3 Prove the following in more ways than one : n+1 FIB(n) (FIB(n+2)=FIB(n+1)+FIB(n); FIB(1)=1,FIB(2)=1. Fibonacci sequence ) Hint PROBLEM 4 We guess that: S M i=1 a i and that; S M i=1 a i kbc are divisible by b c i are relatively prime with respect to b and c). Let : F(n)= M i=1 a i 1+(b-1)(c-1)n Prove by induction that : F(n) is divisible by b c ,for all integers n Solution PROBLEM 5 Let a, b, c be three positive integers where c= a + b . Let p be an odd factor of a +b +c (a +b +c ) is divisible by p.

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