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 ?
Sylvester's work led from the theory of elimination to the theory of invariants and the theory of numbers but he was denied educational and career opportunities because he would not renounce the faith of his birth.
We owe to him the undertaking of innovative research of the highest degree not only in all the branches of geometry, but also in arithmetic and the Euclidean theory of numbers, and these are precisely the domains that would occupy him the most.
With greatest esteem to Leopold Kronecker, one of the founders of the contemporary theory of numbers, it is impossible to agree with him in both the divine origin of number and Man's creation of mathematics.
One feature of Wittgenstein's anti-Platonism is his rejection of classes and the "operational" theory of numbers presented in the Tractatus.
But it is speculative theory, and especially the theory of numbers, that is at the heart of this book.
The p -adic theory of numbers was considered precious to explore many applications in mathematics and computer science since ages.
Shapiro, Introduction to the theory of numbers, John Wiley and Sons, 1983.
Recreations in the Theory of Numbers, Dover Publications, (1966), ISBN 0-486-210960.
Early works of analysis include such topics as the binomial theorem and as a consequence from it the polynomial theorem, purely analytic proof of the theorem and three problems of rectification, complanation and cubature ("as a sample of a complete organization of the science of space"), with later analysis and the infinite represented by pure theory of numbers, theory of functions and improvements, and paradoxes of the infinite.
Another worry I have is that Resnik's theory of the natural numbers as structureless positions in a one dimensional lattice obscures important differences between geometry and the theory of numbers.

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