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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 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. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.

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Elementary number theory

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Number Theory Web - Things of interest to number theorists collected by Keith Matthews.

404 A Mechanical Proof of Quadratic Reciprocity - A paper by David M. Russinoff describing the use of the Boyer-Moore theorem prover in mechanically generating a proof of the Law of Quadratic Reciprocity. PS/PDF.

Active Elementary Number Theory - Interactive (Javascript) expression of a number as a sum of two squares.
Meta Description: [ transformations of relatively prime pairs, representation as sum of two squares, divisors of n*n+1 ]

Covers, Sumsets and Zero-sums - A unified approach to covering systems, restricted sumsets and zero-sum problems by Zhi-Wei Sun.

Fermat's Little Theorem - With notes on Carmichael numbers and the life of R.D. Carmichael.

Frequently Asked Questions - Number Theory section of the sci.math FAQ list.
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Hakmem Continued Fractions - Some notes from the MIT collection. Includes Gosper's algorithms for CF arithmetic.

Infinite Series Theorem - Addresses the question whether a series of rational functions converges to a rational number.

Integer Relations - To determine linear integer dependence among numerical constants and to determine the minimal polynomial of an approximate algebraic number. Interactive or via email.

Introduction to Bernoulli Numbers - A web article with a brief history and account of their relationship with the Riemann zeta function and Fermat's Last Theorem (HTML/PS).

404 Klein Polyhedra - Examples and algorithms for computing Klein polyhedra, also known as Arnold sails or veils (voiles), by Keith Briggs.

L-Functions - Classified by degree, then by level, then by eigenvalue, then by root number.
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Lehmer's Conjecture - That the Mahler measure of an algebraic number is bounded away from 1. Pages by Michael Mossinghoff, UCLA.

Linear Independence Measures for Logarithms of Algebraic Numbers - Michel Waldschmidt. Cetraro summer school lectures, July 2000.

MathPages: Number Theory - Kevin Brown's collection of sci.math postings related to number theory topics.

MathWorld Number Theory - Index to articles in Eric Weisstein's MathWorld in the area of number theory.

Noncommutative Geometry, Trace Formulae and the Zeroes of the Riemann Zeta Function - Lecture notes by Alain Connes.

Number Theory - Dave Rusin's guide to number theory.

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Number Theory Foundation - Aims to collect donations from supporters of number theory and to disburse these donations to encourage research.

Number Theory Glossary - Compiled by Robert Campbell.

On-Line Encyclopedia of Integer Sequences - Given an integer sequence, find its name, and formula.

Online Seminar: Galois theory of p-Extensions - Email seminar on Galoissche Theorie der p-Erweiterungen by Helmut Koch.

Other Maths - A wide selection of number theory topics compiled by Jon Perry.

Some Highlights of Arithmetic Combinatorics - Lecture notes and resources on combinatorial number theory by Terence Tao.

Some Number-Theoretical Constants - Products of rational functions of p over primes, computed by Gerhard Niklasch and Pieter Moree.

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The Arithmetic Properties of Binomial Coefficients - Activated text by Andrew Granville.

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Transcendental Numbers - Maple worksheets, lecture notes and links to other resources by John Cosgrove.

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