number theory


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Related to number theory: Algebraic number theory

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.

Bibliography

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]
(mathematics)
The study of integers and relations between them.
References in periodicals archive ?
Perez, Florentin Smarandache definitions, solved and unsolved problems, conjectures and theorems in Number theory and Geometry, Chicago, Xiquan Publishing House, 2000.
They also show the large gap, particularly in number theory, that existed between Fermat and other mathematicians of his day.
Wiles cracked the Fermat case by first solving a related math riddle central to number theory, the Taniyama Conjecture, which deals with elliptical curves.
Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications.
The field - initiated by Poincare in the study of the N-body problem - has become essential in the understanding of seemingly far off fields such as combinatorics, number theory and theoretical computer science.
The book can be used in a number theory course at the undergraduate or even advanced high school level, and should work for self-study as well.
Number Theory completed a hat-trick at Haydock in July and also races off a 23lb higher mark compared to the first of those wins.
Not always buried deep; a second course in elementary number theory.
Apostol, Introduction to Analytical Number Theory, Spring-Verlag, New York, 1976.
JMD" is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with an emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including number theory, differential geometry, and rigidity, among others.
Ash and Gross describe current research in number theory and explain how the rules of mathematics lead to proofs such as that for Fermat's last theorem.
The domain of number theory lends itself particularly well to generic argument, presented with the intention of conveying the force and structure of a conventional generalized argument through the medium of a particular case.