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→Special-purpose: Added link to the "Difference of two squares" wiki page. The difference of two squares is another popular method for factoring composite integers. |
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In [[number theory]], '''integer factorization''' is the decomposition of a [[positive integer]] into a [[Product (mathematics)|product]] of integers. Every positive integer greater than 1 is either the product of two or more integer [[divisor|factors]], in which case it is called a [[composite number]], or it is not, in which case it is called a [[prime number]]. For example, {{math|15}} is a composite number because {{math|1=15 = 3 · 5}}, but {{math|7}} is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example {{math|1=60 = 3 · 20 = 3 · (5 · 4)}}. Continuing this process until every factor is prime is called '''prime factorization'''; the result is always unique up to the order of the factors by the [[prime factorization theorem]]. A prime factorization algorithm typically involves [[primality test|testing whether each factor is prime]] after each step.
To factorize a small integer {{mvar|n}} using mental or pen-and-paper arithmetic, the simplest method is [[trial division]]: checking if the number is divisible by prime numbers {{math|2}}, {{math|3}}, {{math|5}}, and so on, up to the [[square root]] of {{mvar|n}}. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient.
When the numbers are sufficiently large, no efficient [[quantum computer|non-quantum]] integer factorization [[algorithm]] is known. However, it has not been proven that such an algorithm does not exist. The presumed [[Computational hardness assumption|difficulty]] of this problem is important for the algorithms used in [[cryptography]] such as [[RSA (cryptosystem)|RSA public-key encryption]] and the [[Digital Signature Algorithm|RSA digital signature]].<ref>{{Citation |last=Lenstra |first=Arjen K. |title=Integer Factoring |date=2011 |url=http://link.springer.com/10.1007/978-1-4419-5906-5_455 |encyclopedia=Encyclopedia of Cryptography and Security |pages=611–618 |editor-last=van Tilborg |editor-first=Henk C. A. |place=Boston, MA |publisher=Springer US |language=en |doi=10.1007/978-1-4419-5906-5_455 |isbn=978-1-4419-5905-8 |access-date=2022-06-22 |editor2-last=Jajodia |editor2-first=Sushil}}</ref> Many areas of [[mathematics]] and [[computer science]] have been brought to bear on the problem, including [[elliptic curve]]s, [[algebraic number theory]], and [[quantum computer|quantum computing]].
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