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# Topic: Number theory

 Number theory - Wikipedia, the free encyclopedia Originally, number theory is the branch of pure mathematics concerned with the properties of integers. In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. Number theory was a favorite study among the Ancient Greeks, who were aware of the Diophantine equation concept in numerous special cases. en.wikipedia.org /wiki/Number_theory   (1464 words)

 11: Number theory Number theory is one of the oldest branches of pure mathematics, and one of the largest. For example, "additive number theory" asks about ways of expressing an integer N as 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 of the a_i. 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. The Columbia Encyclopedia, Sixth Edition. 2001-05 branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). An important area in number theory is the analysis of prime numbers. Analytic number theory has given a further refinement of Euclid’s theorem by determining a function that measures how densely the primes are distributed among all integers. www.bartleby.com /65/nu/numberth.html   (334 words)

 Number Theory at the University of Natural Resources (Vienna)   (Site not responding. Last check: 2007-10-21) In fact, since the early 1960's the mathematics chair was held by Karl Prachar whose contributions to the theory of primes (in particular his monograph Primzahlverteilung) obtained the highest esteem by mathematicians all over the world. His research concentrates on the theory of lattice points in large regions and on the asymptotic theory of arithmetic functions. Proceedings of the Number Theory Conference held at Graz in September 1998, eds. www.boku.ac.at /math/nth.html   (807 words)

 Category:Number theory - Wikipedia, the free encyclopedia The main article for this category is Number theory. Traditionally, number theory mathematics concerned with the properties of integers and many open problems that are easily understood even by non-mathematicians. This category roughly corresponds to MSC 11 Number Theory. en.wikipedia.org /wiki/Category:Number_theory   (134 words)

 Number Theory - Numericana Recall that a number is divisible by 3 or 9 iff (if and only if) the sum of its digits is. It is divisible by 11 iff the difference between the sum of its odd digits (units, hundreds, etc.) and the sum of its even digits (tens, thousands, etc.) is so divisible. Since the number 9N divides the number which consists of P nines followed by a certain number J of zeroes, N divides the number consisting of P ones followed by J zeroes, and also the integer composed of P sevens followed by J zeroes. The next two numbers in the list, the 13th and 14th Mersenne primes, are much larger (corresponding to n=521 and n=607) and were both discovered the same day (January 30, 1952, around 22:00 PST and shortly before midnight) by Raphael Mitchel Robinson (1911-1995), at the dawn of the computer age. home.att.net /~numericana/answer/numbers.htm   (7454 words)

 Number Theory   (Site not responding. Last check: 2007-10-21) This number theory seminar also enjoys the active participation of some of the leading figures who come to Montreal on a regular basis and give short courses suitable for graduate students. The number theory group teaches on a regular basis fundamental courses in number theory, algebra and algebraic geometry. Students specializing in number theory are expected to fulfil first the basic requirements in algebra and analysis. www.math.mcgill.ca /department/numtheory.php   (445 words)

 Number Theory Glossary A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases. A Galois Field is a field with finite number of elements. A prime number is a number which has no factors other than 1 (called non-trivial factors). www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html   (827 words)

 Number Theory   (Site not responding. Last check: 2007-10-21) 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]. In multiplicative number theory the connection between squarefree values of polynomials and the abc-conjecture has been studied in [6], while some estimates for pseudo-squares have been given in [11]. In algebraic number theory exponential congruences have been studied in [5], [12] and [18], while a certain problem of I. Korec concerning algebraic integers has been solved in [9]. www.impan.gov.pl /About/numberth.html   (509 words)

 The Math Forum - Math Library - Number Theory   (Site not responding. Last check: 2007-10-21) 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 bibliography of Bernoulli numbers by Karl Dilcher and Ilja Sh. mathforum.org /library/topics/number_theory   (2144 words)

 number theory --  Encyclopædia Britannica the number of atoms, ions, or molecules that a central atom or ion holds as its nearest neighbours in a complex or coordination compound or in a crystal. The next milestone was the atomic theory, advanced in 1805 by an English schoolteacher, John Dalton. This theory states that matter is made up of small particles called atoms, that each chemical element has its own kind of atoms (in contrast to earlier ideas that... www.britannica.com /eb/article-9109430   (867 words)

 Basic Library List-Number Theory Manin, Yuri and Panchishkin, A. Number Theory: Introduction to Number Theory. An Introduction to the Analytic Theory of Numbers. Cassels, J.W. An Introduction to the Geometry of Numbers. www.maa.org /BLL/numtheory.htm   (793 words)

 Number Theory Number Theory at the Mathematics Dept. of the University of Texas John Tate (tate@math.utexas.edu): Algebraic Number Theory (local and global fields), Class Field Theory, Galois cohomology, Galois representations, L-functions and their special values, modular forms, elliptic curves and abelian varieties. Jeffrey Vaaler (vaaler@math.utexas.edu): Analytic number theory, Diophantine approximation and the geometry of numbers in local and global fields, Diophantine inequalities, polynomials, effective measures of irrationality and transcendence, applications of Fourier analysis in number theory, inequalities and extremal problems. www.ma.utexas.edu /users/voloch/numberthy.html   (199 words)

 number theory on Encyclopedia.com   (Site not responding. Last check: 2007-10-21) NUMBER THEORY [number theory] branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). Index number theory using differences rather than ratios. By the numbers: the use of ratings data in academic research. www.encyclopedia.com /html/n1/numberth.asp   (1144 words)

 Explicit algebraic number theory   (Site not responding. Last check: 2007-10-21) The title Explicit algebraic number theory is borrowed from the series of Oberwolfach meetings on Explicit methods in number theory. The advanced techniques from algebraic number theory that apply to these problems include class field theory, infinite Galois theory, and the theory of quadratic forms. The purpose of this part is to impart a working knowledge of these theories to the participants, to provide ample illustrations of their use, and to formulate several open problems that may be approachable by means of the same techniques. www.math.leidenuniv.nl /~psh/EANT   (334 words)

 Number Theory - Mathematics and the Liberal Arts When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today. The author discusses parameterization of Pythagorean triangles, the law of quadratic reciprocity, representation of numbers in a fixed finite number of sums of squares numbers, quadratic forms, and connections with the complex numbers, quaternions, and Cayley numbers. math.truman.edu /~thammond/history/NumberTheory.html   (1152 words)

 Math - Number Theory Aims to collect donations from supporters of number theory and to disburse these donations to encourage research. That the Mahler measure of an algebraic number is bounded away from 1. To determine linear integer dependence among numerical constants and to determine the minimal polynomial of an approximate algebraic number. www.canadiancontent.net /dir/Top/Science/Math/Number_Theory   (481 words)

 Computational number theory 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 six largest numbers factored so far with NFS have 174 (factored on December 3, 2003), 160 (factored on April 1, 2003), 158 (factored on January 19, 2002), 155, 140, and 130 decimal digits. 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)

 Algebraic Number Theory Archive   (Site not responding. Last check: 2007-10-21) This archive is for research in algebraic number theory and arithmetic geometry. ANT-0296: 8 Jun 2001, On the Iwasawa theory of p-adic Lie extensions, by Otmar Venjakob. ANT-0267: 27 Nov 2000, On an analogue for number fields of a conjecture of de Jong on F_q[[t]]-analytic extensions of function fields, by Gebhard Boeckle. front.math.ucdavis.edu /ANT   (12251 words)

 Math 254 (Number Theory)   (Site not responding. Last check: 2007-10-21) This was the official course web page for Math 254B (Number Theory) at UC Berkeley, which I taught during the Spring 2002 semester. The Berkeley Number Theory Seminar meets Wednesdays from 3:10 to 4:00 PM in 891 Evans, and sometimes on Friday at the same time and place. Prospective students in number theory are encouraged to attend. www-math.mit.edu /~kedlaya/math254b.html   (400 words)

 Research in Number Theory & Combinatorics Both number theory and combinatorics are part of what is called discrete mathematics, which has important applications in computer science and information technology, as well as an intrinsic elegance and fascination for mathematicians, professionals and amateurs alike. Number theory originated as the study of the structure and properties of the ordinary integers, but nowadays has expanded into the study of analogous properties of other (possibly non-commutative) rings. Current topics of interest in the department include: combinatorial design theory; automorphism groups of graphs and designs; Hadamard matrices and symmetric designs and their classification; applications of combinatorics to computer graphics. www.maths.gla.ac.uk /research/groups/ntc/ntc.html   (256 words)

 Number Theory Two classic essays by great German mathematician: on the theory of irrational numbers; and on transfinite numbers and properties of natural numbers. Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields. Gausian primes; polynomials over a field; algebraic number fields; algebraic integers and integral bases; uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal c... store.doverpublications.com /0486682528.html   (383 words)

 Fields Institute - Conference in Number Theory - 2003 Conference in Number Theory in Honour of Professor H.C. Williams 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. www.fields.utoronto.ca /programs/scientific/02-03/numtheory   (409 words)

 Elementary Math from Math Goodies including exponents, factors, multiples, prime, composite, divisibility To determine if a number is prime or composite by examining its factors. To determine if larger whole numbers are prime or composite by using tests for divisibility. To predict the next number in a sequence of numbers written in exponential form. www.mathgoodies.com /lessons/toc_vol3.html   (252 words)

 Earliest Uses of Symbols of Number Theory Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. I think the first Bourbaki volume published was the results summary on set theory, in 1939, and it does not contain any symbol for the complex numbers. The symbol φ(m) for the number of integers less than m that are relatively prime to m was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in his Disquisitiones arithmeticae articles 38, 39 (p. members.aol.com /jeff570/nth.html   (1263 words)

 London Number Theory Seminar   (Site not responding. Last check: 2007-10-21) The London Number Theory Seminar is held weekly during term times. This term, the seminar will be hosted by King's College and will be held from 4 to 5 pm, in room 423 of the Mathematics department, located on the 4th floor of the Strand Building. In one case the Riemann Hypothesis of Deligne is the crucial ingredient, in the other the spectral theory of automorphic forms appears naturally. www.mth.kcl.ac.uk /events/numbtheo.html   (502 words)

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