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In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [3] [4] [5] For example,
The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Some of these are classification theorems of objects which are mainly dealt with in the field.
Fundamental theorem of algebra. The fundamental theorem of algebra, also called d'Alembert's theorem[ 1] or the d'Alembert–Gauss theorem, [ 2] states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a ...
Influential theorems in number theory include the fundamental theorem of arithmetic, Euclid's theorem, and Fermat's last theorem. [89] According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers.
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm (a, b), is the smallest positive integer that is divisible by both a and b. [1] [2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both ...
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: [note 1] Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b . For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and since this ...
Bézout's identity. In mathematics, Bézout's identity (also called Bézout's lemma ), named after Étienne Bézout who proved it for polynomials, is the following theorem : Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form ...
Paul Erdős gave a proof [11] that also relies on the fundamental theorem of arithmetic. Every positive integer has a unique factorization into a square-free number r and a square number s 2. For example, 75,600 = 2 4 3 3 5 2 7 1 = 21 ⋅ 60 2.