number theory, 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. A prime number is an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p1 · p2 · p3 · · · pn) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. 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. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard Euler, C. F. Gauss, and Pierre de Fermat. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.
See O. Ore, Number Theory and Its History (1988); R. P. Burn, A Pathway into Number Theory (2d ed. 1996); J. H. Silverman, A Friendly Introduction to Number Theory (1996); M. A. Herkommer, Number Theory: A Programmer's Guide (1998); R. A. Mollin, Algebraic Number Theory (1999).
Questia, a part of Gale, Cengage Learning. www.questia.com
Publication information: Article title: number theory. Encyclopedia title: The Columbia Encyclopedia, 6th ed.. © 2012 The Columbia Electronic Encyclopedia © 2012, Columbia University Press. Licensed from Columbia University Press. Used with the permission of Columbia University Press. All Rights Reserved. Publisher: The Columbia University Press. Place of publication: Not available. Publication year: 2013.
This material is protected by copyright and, with the exception of fair use, may not be further copied, distributed or transmitted in any form or by any means.