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...odular arithmetic]], having the same remainder when divided by a specified integer ...oup defined by congruence conditions on the entries of a matrix group with integer entries ...

...lished by [[Springer Science+Business Media|Springer]] as volume 19 of the Algorithms and Computations in Mathematics series.<ref>{{Cite web|url=https://magma.ma ...[[Integer factorization]] algorithms include the [[Lenstra elliptic curve factorization|Elliptic Curve Method]], the [[Quadratic sieve]] and the [[General number f ...

{{unsolved|computer science|Can integer factorization be solved in polynomial time on a classical computer?}} ...the result is always unique up to the order of the factors by the [[prime factorization theorem]]. ...

...lly denoted by {{nowrap|lcm(''a'',&nbsp;''b'')}}, is the smallest positive integer that is [[divisible]] by both ''a'' and ''b''.<ref name=":1">{{Cite web|las ...a'',&nbsp;''b'',&nbsp;''c'', . . .)}}, is defined as the smallest positive integer that is divisible by each of ''a'', ''b'', ''c'', . . .<ref name=":1" /> ...

with [[integer]] coefficients <math>a_i\in\mathbb{Z}</math> and <math>a_0,a_n \neq 0</math * {{math|''p''}} is an integer [[divisor|factor]] of the [[constant term]] {{math|''a''₀}}, and ...

...sor|divisible]] by no [[square number]] other than 1. That is, its [[prime factorization]] has exactly one factor for each prime that appears in it. For example, {{ ==Square-free factorization== ...

...'' of {{math|15}}, and {{math|(''x'' − 2)(''x'' + 2)}} is a ''[[polynomial factorization]]'' of {{math|''x''² − 4}}. ...mes(1/y)</math> whenever <math>y</math> is not zero. However, a meaningful factorization for a [[rational number]] or a [[rational function]] can be obtained by wri ...

...ithm''' is an [[algorithm]] designed to solve a [[search problem]]. Search algorithms work to retrieve information stored within particular [[data structure]], o Although [[Search engine (computing)|search engines]] use search algorithms, they belong to the study of [[information retrieval]], not algorithmics. ...

...s whether the guess is a solution to the problem.<ref>Alsuwaiyel, M. H.: ''Algorithms: Design Techniques and Analysis'', [https://books.google.com/books?id=SPx4i Assume that we are given some [[integer]]s, {−7, −3, −2, 5, 8}, and we wish to know whether some of these integers ...

...putational number theory]], especially [[primality testing]] and [[integer factorization]]; these in turn are important in [[cryptography]]. For any integer ''a'' and any positive odd integer ''n'', the Jacobi symbol {{big|(}}{{sfrac|''a''|''n''}}{{big|)}} is defined ...

...rix]]. Some variants are commonly referred to as '''square-and-multiply''' algorithms or '''binary exponentiation'''. These can be of quite general use, for exam The method is based on the observation that, for any integer <math>n > 0</math>, one has ...

...ath>p</math> is an odd [[prime number]] and <math>a</math> is a positive [[integer]] that may or may not be a [[quadratic residue]] mod&nbsp;''p''. The Legen ...ovides an efficient way to compute all Legendre symbols without performing factorization along the way. ...

...on]] which allows a suitable generalization of [[Euclidean division]] of [[integer]]s. This generalized Euclidean algorithm can be put to many of the same us of them ([[Bézout's identity]]). In particular, the existence of efficient algorithms for Euclidean division of integers and of [[polynomial]]s in one variable o ...

{{short description|Computation modulo a fixed integer}} ...], '''modular arithmetic''' is a system of [[arithmetic]] operations for [[integer]]s, differing from the usual ones in that numbers "wrap around" when reachi ...

...of superpolynomial speedup compared to best known classical (non-quantum) algorithms.<ref name=":0" /> However, beating classical computers may require quantum ...ually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances ...

...he fastest known attack against an algorithm), because the security of all algorithms can be violated by [[brute-force attack]]s. Ideally, the lower-bound on an ...ided that there is no analytic attack (i.e. a "structural weakness" in the algorithms or protocols used), and assuming that the key is not otherwise available (s ...

...[[prime power]]. For every prime number <math>p</math> and every positive integer <math>k</math> there are fields of order <math>p^k</math>. All finite field ...> (where <math>p</math> is a prime number and <math>k</math> is a positive integer). In a field of order <math>p^k</math>, summing <math>p</math> copies of an ...

...est means of approaching this system is direct solution (for example, [[LU factorization]]), which for small problems is very practical. For large problems, the sys ...{Cite book |last1=Kapoor |first1=S |last2=Vaidya |first2=P M |chapter=Fast algorithms for convex quadratic programming and multicommodity flows |date=1986-11-01 ...

To find all the prime numbers less than or equal to a given integer {{mvar|n}} by Eratosthenes's method: ...rs is probably not by Nicomachus. --> This can be generalized with [[wheel factorization]], forming the initial list only from numbers [[coprime]] with the first fe ...

{{Short description|Largest integer that divides given integers}} ...two or more [[integer]]s, which are not all zero, is the largest positive integer that [[divides]] each of the integers. For two integers {{math|''x''}}, {{m ...