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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."
Conventionally, the number of real embeddings of K is denoted r 1, while the number of conjugate pairs of complex embeddings is denoted r 2. The signature of K is the pair (r 1, r 2). It is a theorem that r 1 + 2r 2 = d, where d is the degree of K.
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.
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 ...
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.
In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {}. In axiomatic set theory , the existence of singletons is a consequence of the axiom of pairing : for any set A , the axiom applied to A and A asserts the existence of { A , A } , {\displaystyle \{A,A\},} which is the same as the ...
The smallest integer m > 1 such that p n # + m is a prime number, where the primorial p n # is the product of the first n prime numbers. A005235: Semiperfect numbers: