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	Number theory - Wikipedia, the free encyclopedia
		Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers.
		Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated.
		In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
	en.wikipedia.org /wiki/Number_theory (1354 words)

	11: Number theory
		Number theory is one of the oldest branches of pure mathematics, and one of the largest.
		Elementary number theory involves divisibility among integers -- the division "algorithm", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets Z/nZ as commutative rings), including Fermat's little theorem and Euler's theorem extending it.
		Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions).
	www.math.niu.edu /~rusin/known-math/index/11-XX.html (2572 words)

	Number Theory (Site not responding. Last check: 2007-10-18)
		The number theory group at Oklahoma State University has established a thriving program of research, including a regular seminar series featuring lectures of both a research and expository nature by the resident number theorists, as well as frequent lectures by distinguished young and senior number theorists from around the country.
		Number theory is famed not just for the beauty of its theorems, but for the enormous wealth and variety of techniques involved in discovering and proving these theorems.
		Our faculty is prepared to offer courses in algebraic number theory, class field theory, analytic number theory, the arithmetic of elliptic curves as well as other arithmetic algebraic varieties, p-adic analysis, automorphic and modular forms, discrete subgroups of algebraic groups, computational number theory, as well as many other subfields of number theory.
	mathgrad.cas.okstate.edu /handbook/node53.html (405 words)

	Computational Number Theory (Site not responding. Last check: 2007-10-18)
		Computational Number Theory (CNT) may loosely be defined as the use of computational techniques to answer questions in Number Theory, or to gain insights into how questions may be answered by a theoretical approach.
		Number Theory itself may be described as the mathematical treatment of the properties and relationships between integers.
		Number Theory used to be completely useless but, since the invention of public key cryptography in the 1970's, it's now reached the stage where small sums of money can be earned by factoring large integers, and extremely large sums of money are protected by integers which are supposedly impossible to factor.
	www.leyland.vispa.com /numth/CNT.htm (144 words)

	Computational Science, Directory (Site not responding. Last check: 2007-10-18)
		Computational Projects in Number Theory Interesting problems, usually requiring extensive verifications or enumerations, to occupy the idle CPU time of workstations and personal computers.
		Future Directions in Algorithmic Number Theory Some of the conjectures and open problems motivated by the "PRIMES is in P" papers, compiled at the AIM.
		CLINT Hugh Montgomery's Computational Laboratories in Number Theory.
	www.morrisarearedcross.org /bWFfNzc0MzY5.aspx (168 words)

	Computational number theory
		Computational number theory studies problems from elementary, algebraic and analytic number theory which require the help of fast computers, particularly vector and parallel systems.
		For example, some algorithms for factoring large numbers can be carried out on a grid of heterogeneous computers where the number of computers in the grid is allowed to vary in a dynamical way.
		The number and the size of known amicable number pairs has grown explosively in recent years: from 1108 amicable pairs in 1972 (the largest pair consisting of two 25-digit numbers), to more than five million amicable pairs in May 2003 (the largest pair consisting of two 5577-digit numbers).
	db.cwi.nl /projecten/project.php4?prjnr=84 (1003 words)

	Computational Number Theory
		Here are some useful routines for making computations with positive definite binary quadratic forms: computing representatives of class group, class number, composition of forms, etc. To use it download the file bforms.gp and then check the file bforms.txt for details and examples.
		This routine compute a Weierstrass form for the Jacobian of a general homogeneous cubic in 3 variables (over a field of arbitrary characteristic); it is joint work with John Tate.
		Computation of Dedekind's eta function on CM points using its modular properties to relate the value to that of the corresponding point in the standard fundamental domain.
	www.ma.utexas.edu /users/villegas/cnt/cnt-no-frames.html (566 words)

	Read This: A Course in Computational Number Theory
		Nonetheless, CNT emphasizes some parts of elementary number theory which are really the beginning of large algebraic and analytic theories.
		A virtue of CNT is that it is truly aimed at the entire pool, not just the much smaller collection of students who are thinking about going on to graduate school in mathematics.
		One way CNT is aimed at the entire pool is that it uses the common language of mathematical science, not the specialized language of pure mathematics.
	www.maa.org /reviews/bresswagon.html (1036 words)

	Basics of Computational Number Theory (Site not responding. Last check: 2007-10-18)
		This document is a gentle introduction to computational number theory.
		When computing a power of a number with a finite modulus there are efficient ways to do it and inefficient ways to do it.
		A number which is not prime is said to be composite.
	www.math.umbc.edu /~campbell/NumbThy/Class/BasicNumbThy.html (2263 words)

	4070/5070 -- Computational Number Theory (Site not responding. Last check: 2007-10-18)
		In order to find rules explaining the ''behaviour'' of numbers, mathematicians have in all times used sample computations, first carried out by hand, later with the help of devices such as mechanical calculators and computers.
		Number theory has particularly profited from these developments, leading to its own branch known as Computational Number Theory.
		This new course is addressed to both mathematics students interested in computing/computer science as well as to computer science students with interests in mathematics and number theory in particular.
	www.mscs.dal.ca /~joerg/crs/4-5070.html (401 words)

	Computational
		CLINT - Hugh Montgomery's Computational Laboratories in Number Theory.
		Computational Projects in Number Theory - Interesting problems, usually requiring extensive verifications or enumerations, to occupy the idle CPU time of workstations and personal computers.
		Future Directions in Algorithmic Number Theory - Some of the conjectures and open problems motivated by the "PRIMES is in P" papers, compiled at the AIM.
	www.supercrawler.com /Science/Math/Number_Theory/Computational (272 words)

	Amazon.com: A Course in Computational Algebraic Number Theory (Graduate Texts in Mathematics): Books (Site not responding. Last check: 2007-10-18)
		It contains descriptions of 148 algorithms, which are fundamental for number theoretic calculations, in particular for computations related to algebraic number theory, elliptic curves, primality testing, lattices and factoring.
		(b) The computation of the coefficient g2 and g3 of the Weierstrass equation of an elliptic curve.
		The author gives an overview of the computer packages used for number theory, including Pari, which was written by him and his collaborators.
	www.amazon.com /exec/obidos/tg/detail/-/0387556400?v=glance (1176 words)

	The Math Forum - Math Library - Number Theory
		Papers from a Mathematics graduate from The University Of Sussex at Brighton: Number Theory: GCD and Prime Factorisation; Molien's Theorem, Invariant Theory and Gregor Kemper; A History of Equality.
		In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer.
		A connected series of four problems in elementary number theory that are ideal for discovery learning at several levels.
	mathforum.org /library/topics/number_theory (2145 words)

	Arizona Mathematics \| Research \| Number Theory
		The research of the number theory group encompasses classical and algebraic number theory, computational number theory, and especially the modern subject of arithmetical algebraic geometry.
		The number of Fields Medals (the mathematical equivalent of the Nobel prize) awarded for work in the area is a testament to its centrality in modern fundamental mathematics.
		Two of the Millenium Prize Problems in mathematics, offered by the Clay Mathematics Institute, are in the area of number theory and one more is closely related to number theory.
	math.arizona.edu /research/numbertheory.html (289 words)

	Amin Shokrollahi's Publications (Site not responding. Last check: 2007-10-18)
		A displacement structure approach to decoding algebraic geometric codes, V. Olshevksy and A. Shokrollahi, Proceedings of the 31st annual ACM Symposium on Theory of Computing (STOC), 1999, pp.
		Fast and precise computation of discrete Fourier Transforms using cyclotomic integers, J.P. Buhler, A. Shokrollahi and V. Stemann, Proceedings of the 29th annual ACM Symposium on Theory of Computing (STOC), 1997, pp.
		Deciding properties of number fields without factoring, T. Sander and A. Shokrollahi, Proceedings of the 28th IEEE Symposium on the Foundations of Computer Science (FOCS), 1997, pp.
	www.shokrollahi.com /amin/pub.html (1061 words)

	Powell's Books - Graduate Texts in Mathematics #193: Advanced Topics in Computational Number Theory by Henri Cohen
		Elementary analysis :the theory of calculus / Kenneth A. Ross.
		Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms.
		The highlights are the algorithms for computing the structure of (Z_K/m)x, of ray class groups, and relative equations for Abelian extensions based on complex multiplication of Stark's conjectures.
	www.powells.com /cgi-bin/biblio?isbn=0387987274 (514 words)

	Computational Number Theory
		With this practical concern in mind, they’re engaged in serious study of computational number theory: thinking of new ways, new mathematical tricks to speed up the processes of encryption, decryption, and key generation and make them more secure and more efficient.
		Today’s ordinary computer can do it in a fraction of a second, but on the Internet, even this can be too slow.
		Montgomery was a member of the team that recently succeeded in factoring a number of 155 digits — it took them four months with a roomful of powerful computers.
	research.microsoft.com /research/crypto/compnum.asp (489 words)

	Computational number theory
		Contributions (in the form of factored numbers) to the international Magma project.
		CWI has several source code license agreements with companies in The Netherlands, USA, Germany and France which allow them to use the Number Field Sieve factorization code as this was and is being developed by P.L. Montgomery, A.K. Lenstra, M.
		A group of the Royal Institute of Technology in Stockholm (Hastad) has used this code as a basis for factoring a hard 512-bit challenge number in October 2000.
	db.cwi.nl /projecten/project.php4?prjnr=84&type=deliv (109 words)

	American Mathematical Monthly, The: A Computational Introduction / A Course in Computational Number Theory (Site not responding. Last check: 2007-10-18)
		Together they represent almost opposite ends of the spectrum in the use of computer algebra tools for pedagogical purposes and thus are natural foils for each other.
		Next is a chapter on Galois theory, followed by a chapter on quartic equations and their Galois groups and the resolvent cubics.
		Scherk's book is a rather nice treatment of abstract algebra, but I think it fails in its attempt to introduce computation because it separates the mathematics and the computation.
	www.findarticles.com /p/articles/mi_qa3742/is_200201/ai_n9046371 (644 words)

	The USC Number Theory Home Page
		I study modular forms and their applications to problems relating to algebraic number theory, elliptic curves, L-functions, partitions, and other topics in number theory.
		Analytic Number Theory and Approximation Theory with particular interests in the use of finite differences to determine information about lattice points close to a curve or surface.
		Analytic and Elementary Number Theory with particular interests in the distribution of primes, Waring's problem, arithmetic properties of elliptic curves over the rationals, and applications of the theory of modular forms.
	www.math.sc.edu /~filaseta/numthry.html (398 words)

	Computational Number Theory at CWI in 1970-1994 (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		Computational Number Theory at CWI in 1970-1994 (1994)
		327 A Course in Computational Algebraic Number Theory (context) - Cohen - 1993 ACM
		50 On distinguishing prime numbers from composite numbers (context) - Adleman, Pomerance et al.
	citeseer.ist.psu.edu /teriele94computational.html (682 words)

	MTH-2D12 : Computational Number Theory (Site not responding. Last check: 2007-10-18)
		Overview: It is not true to say that this is a new subject: in a sense, Number Theory has always had one eye on computational aspects.
		However it is true to say that the subject advanced massively with the advent of electronic computing.
		Recent e-mails to the 'Number Theory Network' on the web will be circulated to keep students up-to-date with current progress on these topics.
	www.mth.uea.ac.uk /maths/syllabuses/9900/2D1200.html (459 words)

	Magma Computational Algebra System Home Page
		Magma is a large, well-supported software package designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics.
		October 18, 2004: The Algebraic Geometry and Number Theory with Magma conference was held October 4 - 8, 2004 at the Institute Henri Poincaré, Paris.
		Magma is produced and distributed by the Computational Algebra Group within the School of Mathematics and Statistics of the University of Sydney.
	magma.maths.usyd.edu.au /magma (236 words)

	DESCRIPTIONS OF AREAS/COURSES IN NUMBER THEORY, LECTURE NOTES (Site not responding. Last check: 2007-10-18)
		Analytic Number Theory and Applications: Collection of papers on the occasion of the 60th birthday of Anatolli Alexeevich Karatsuba, Proc.
		Computational class field theory, A course given at the Middle East Technical University, Ankara, 1997 by Henri Cohen (dvi 255K)
		An Index for G.H. Hardy and E.M Wright: An Introduction to the theory of numbers
	www.numbertheory.org /ntw/N4.html (1977 words)

	BUBL LINK: Number theory
		Specific areas of concentration include structure of complexity classes, algebraic complexity, cryptography, interactive proofs, complexity issues in: computational geometry, robotics, and motion planning, learning theory, number theory, logic, combinatorial optimisation and approximate solutions, and distributed computing.
		Mathematics of Computation is devoted to research articles in computational mathematics.
		Areas covered include numerical analysis, with emphasis on the mathematical analysis and development of methods, computational number theory and algebra, and related fields.
	www.bubl.ac.uk /link/n/numbertheory.htm (272 words)

	Number theory files for David Eppstein
		Conway's nimbers (used in combinatorial game theory) form an infinite field of characteristic two, with a natural binary representation in which truncation to a fixed number of bits produces finite subfields GF[2^2^k].
		The p-adic numbers, much beloved of a certain net.crackpot, can be thought of as describing arithmetic modulo powers of p^k, in the limit as k becomes large.
		If one interprets (x,y) as the complex number (Gaussian integer) x+iy, the form (x^2+y^2) is its norm, and we can form similar pictures by using norms of other rings.
	www.ics.uci.edu /~eppstein/numth (464 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Algorithmic Number Theory These notes were written as part of a seminar that I ran during Fall 2000 at CSUN.
		Algorithmic Number Theory I Cyclicity of units of Z/p^n for odd p, Carmichael numbers.
		Algorithmic Number Theory IV Factorization: Pollard's p-1 test, and Lenstra's elliptic curve algorithm.
	www.csun.edu /~asethura/notes.html (133 words)

	position "Computational Number Theory and Data Security" (Site not responding. Last check: 2007-10-18)
		The research group "Computational Number Theory and Data Security" at CWI has a vacancy for a
		For the validation of RSA, the aim is to study, improve and analyse existing algorithms, and develop new algorithms for factoring the largest possible RSA keys.
		The candidate is expected to have a Master's degree in mathematics and/or computer science with emphasis on cryptography, number theory, and programming.
	www.win.tue.nl /math/eidma/positioncntds.html (249 words)

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