number theory

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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 EulerEuler, Leonhard
, 1707–83, Swiss mathematician. Born and educated at Basel, where he knew the Bernoullis, he went to St. Petersburg (1727) at the invitation of Catherine I, becoming professor of mathematics there on the departure of Daniel Bernoulli (1733).
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, C. F. GaussGauss, Carl Friedrich
, born Johann Friederich Carl Gauss, 1777–1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ.
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, and Pierre de FermatFermat, Pierre de
, 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
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. 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).

number theory

[′nəm·bər ′thē·ə·rē]
The study of integers and relations between them.
References in periodicals archive ?
There are not too many number theorists who use computers to assist them in proving theorems, though there are more everyday," Hanke says.
To number theorists, partitions are among the most tantalizing objects in mathematics.
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Since then, Catalan's conjecture has posed a challenge to number theorists akin to that provided by Fermat's last theorem (SN: 11/5/94, p.
Almost all number theorists consider the first Fermat prime to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that the first five Fermat numbers are prime.
Factoring has long intrigued and perplexed number theorists.
Number theorists are interested in the specific instances when x and y are both fractions, or rational numbers.
Interestingly, Beal's conjecture is closely related to questions of considerable concern to number theorists.
The discovery of such a large Mersenne prime by a hit-or-miss approach actually has little value for computational number theorists interested in the distribution of prime numbers and related mathematical issues.
Most number theorists, for a variety of convincing reasons, believe the structural conjecture to be true, although it has not been proven.