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Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."
This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. For example, −2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the ...
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1] It is often said to have begun with Peter Gustav Lejeune Dirichlet 's 1837 introduction of Dirichlet L -functions to give the first proof of Dirichlet's theorem on arithmetic progressions .
Prime number: A positive integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A positive integer that can be factored into a product of smaller positive integers. Every integer greater than one is either prime or composite.
1 (one, unit, unity) is a number, numeral, and glyph.It is the first and smallest positive integer of the infinite sequence of natural numbers.This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a ...
If x is an algebraic number then a n x is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where a n x n is the highest-degree term of p(x). The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients.
The word integer comes from the Latin integer meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier, which means both entire and integer. [9] Historically the term was used for a number that was a multiple of 1, [10] [11] or to the whole part of a ...
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem , first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer ...