Integer factorization - Integer factorization In mathematics, the integer prime-factorization (also known as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. For example, given the number 45, the prime factorization would be 32·5. The factorization is always unique, according to the fundamental theorem of arithmetic. This problem is of significance in mathematics, cryptography, complexity theory, and quantum computers. The complete list of factors can be derived from the prime factorization by incrementing the exponents from zero until the number is reached. For example, since 45 = 32·5, 45 is divisible by 30·50, 30·51, 31·50, 31·51, 32·50, and 32·51, or 1, 5, 3, 15, 9, and 45. In contrast, the prime factorization only includes prime factors. Given two large prime numbers,.
Gaussian integer - Gaussian integer A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. This is a Euclidean domain which cannot be turned into an ordered ring. The norm of a Gaussian integer is the natural number defined as N(a + bi) = a2 + b2. The norm is multiplicative, i.e. N(zw) = N(z)N(w). The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements 1, -1, i and -i. The prime elements of Z[i] are also known as Gaussian primes. Some prime numbers are not Gaussian primes; for example 2=(1+i)(1-i) and 5=(2+i)(2-i). Those prime numbers which are congruent to 3.
Factorization - Factorization In mathematics, factorization or factoring is the decomposition of an object into a list of (smaller) objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5; and the polynomial x2 - 4 factors as (x - 2)(x + 2). The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of.
Prime factorization algorithm - Prime factorization algorithm A prime factorization algorithm is an algorithm (a step-by-step process) by which an integer (whole number) is "decomposed" into a product of factors that are prime numbers. The Fundamental Theorem of Arithmetic guarantees that this decomposition is unique. Table of contents showTocToggle("show","hide") 1 A simple factorization algorithm 1.1 Description 1.2 Time complexity 2.
Lenstra elliptic curve factorization - Lenstra elliptic curve factorization The Lenstra elliptic curve factorization is a fast probabilistic algorithm for integer factorization which employs elliptic curves. This method was the best algorithm for integer factorization until the general number field sieve was developed. It is still best for factoring out divisors of size not exceeding 20 digits (64 bits), as its running time depends on the size of a factor p rather than on the size of the number n to be factored. It is an improvement of the older Pollard p −1 factorization method. In that method, it was assumed that the given number n has a prime factor p such that p-1 had only "small" prime factors (numbers whose prime factors are all "small" are informally called smooth). Then, by Fermat's little.
Ideal class group - of unity, was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime). Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that.
Infinite product - formulae for Ï€, such as the following two products, respectively by Viète and Wallis: 2/Ï€ = (√2 / 2)(√(2 + √2) / 2)(√(2 + √(2 + √2)) / 2)... Ï€/2 = (2/1)(2/3)(4/3)(4/5)(6/5)(6/7)(8/7)(8/9)... Product representations of functions One important result concerning infinite products is that every function f(z) which is entire, i.e. holomorphic over the entire complex plane, can be factored into an infinite product of entire functions each with at most a single zero. In general, if f has a zero of order m at the origin and has other complex zeros at u1, u2, u3, ... (listed with multiplicities equal to their orders) then f(z) = zmeφ(z) Î (1 - z/un) exp[z/un + (z/un)2/2 + ... + (z/un)λn] where λn are positive integers that can be chosen to make the.
Integral domain - Examples The prototypical example is the ring Z of all integers. Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields. Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients . The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the.
General number field sieve - known for factoring integers. It uses O(exp( ((64/9) n)1/3 (log n)2/3 )) steps to factor integer n. It is an improvement of older sieving method which factors n by finding numbers ki such that ri=ki2-n factor completely over a fixed set (called basis) of small primes. Then, having enough such ri - which are called smooth relative to the chosen basis of primes, using Gauss elimination method of linear algebra we can choose exponents ci equal to 0 or 1 such that product of rici is a square, say x2. On the other hand, if the product of kici is y, then x2-y2 is divisible by n and with probability at least one half we get a factor of n by finding greatest common divisor of n and x-y. In this.
Unary numeral system - is very cumbersome, however. Compared to positional numeral systems, the unary system is inconvenient and is not used in practice for large calculations. It occurs in some problem descriptions in theoretical computer science (e.g. some P-Complete problems), where it is used to "artificially" decrease the run-time or space requirements of a problem. For instance, the problem of integer factorization is suspected to require more than polynomial run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary..
Glossary of ring theory - 5 Miscellaneous Definition of a ring A ring is an abelian group (R,+) together with an associative operation * which is distributive over + and has an identity element 1 with respect to *. The operation + is referred as the addition and * is referred as the multiplication. The identity element with respect to + is written as 0. ;Characteristic : The characteristic of a ring is the smallest positive integer n satisfying n1=0 if it exists and 0 otherwise. In particular ne=0 for all elements e of the ring. Types of elements ; Idempotent : An element e of a ring is idempotent if e2 = e. ; Central : An element r of a ring R is central if xr = rx for all x in R. The.
Greatest common divisor - (HCF) of two integers which are not both zero is the largest integer that divides both numbers. The GCD of a and b is often written as gcd(a,b) or simply (a,b). For example, gcd(12,18) = 6, gcd(-4,14) = 2 and gcd(5,0) = 5. Two numbers are called coprime or relatively prime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime. The greatest common divisor is useful for writing fractions in lowest terms. Consider for instance 42/56 = 3/4 where we cancelled 14, the greatest common divisor of 42 and 56. Table of contents showTocToggle("show","hide") 1 Calculating the GCD 2 Properties 3 GCD in commutative rings Calculating the GCD While the GCD of two numbers can in principle be computed by determining the prime factorizations of.
Fifteen - a triangular number. It is a composite number; its proper divisors being 1, 3 and 5. Its factorization is In hexadecimal, as well as all higher bases, fifteen is represented as F. Fifteen is the magic result of a 3 by 3 magic square: Fifteen is also: The atomic number of phosphorous. The number of guns in a gun salute to Army, Air Force and Marine Corps Lieutenant Generals, and Navy and Coast Guard Vice Admirals. The designation of Interstate 15, a freeway that runs from California to Montana. The age of a quinceañera, a Hispanic girl celebrating her fifteenth birthday. The age for obtaining a driver's permit in jurisdictions where the age for a driver's license is sixteen. The number of players on a rugby union team. See also: thirteen,.
Fifty - include: Roman numerals: L decimal or Arabic notation: 50 Its factorization is It is the smallest number that can be written as the sum of two squares in two distinct ways: 50 = 12 + 72 = 52 + 52. It is also the sum of three squares, 50 = 32 + 42 + 52. Fifty is also: The atomic number of tin. The number of states in the USA (since 1959, still true in 2004) A calibre of ammunition (0.50 inches: see .50 BMG) In millimeters, the focal length of the normal lens in 35mm photography. the percentage equivalent to one half, so that "fifty-fifty" is commonly used for something divided equally in half, or an event of probability one half In U.S. dollars, the denomination of the Federal Reserve.
Formal power series - rearrangement of the series converges to the same limit. This topological ring is the ring of formal power series over R and is denoted by R[[X]]. Properties R[[X]] is an associative algebra over R which contains the ring R[X] of polynomials over R; the polynomials correspond to the sequences which end in zeros. The geometric series formula is valid in R[[X]]: An element ∑ an Xn of R[[X]] is invertible in R[[X]] if and only if its constant coefficient a0 is invertible in R. This implies that the Jacobson radical of R[[X]] is the ideal generated by X and the Jacobson radical of R. The maximal ideals of R[[X]] all arise from those in R in the following manner: an ideal M of R[[X]] is maximal if and only if M.
Fundamental theorem of arithmetic - particular number theory, the fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. For instance, we can write 6936 = 23 · 3 · 172 or 1200 = 24 · 3 · 52 and there are no other such factorizations of 6936 or 1200 into prime numbers, except for reorderings of the above factors. To make the theorem work even for the number 1, we think of 1 as being the product of zero prime numbers (see empty product). Applications The theorem establishes the importance of prime numbers. Essentially, they are the "basic building blocks" of the positive integers, in that every positive integer can be put together from primes in a unique fashion..
Eighty - Ï€´ Factors: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 Factorization: Ordinal number: eightieth Numeral system base 80: octagesimal Name in other languages: quatre-vingt (in French, from four and twenty), see also wiktionary:eighty See also: seventy, eighty, ninety, integer, list of numbers. 80 is also: the atomic number of mercury (Hg) the number of units in a fourscore the age at which one becomes an octogenarian in the title of Around the World in Eighty Days by Jules Verne the model number of the computers TRS-80, IBM 80, Sinclair ZX80, Aster CT-80 the model number of T-80, a Soviet tank; IAR 80, a Romanian WWII fighter aircraft; McDonnell Douglas MD-80, Arado Ar 80, F-80 Shooting Star part of the name of the bands Eighty-D and Eighty Proof Soul the.
Elliptic curve - theorem and they also find applications in cryptography (for details, see the article elliptic curve cryptography) and integer factorization. These curves are not ellipses: see elliptic integral for the origin of the term. Elliptic curves are non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group. Typical elliptic curves over the field of real numbers are given by the equations y2 = x3 − x and y2 = x3 − x + 1. Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1. If the characteristic of K is neither 2.
Exponential time - Exponential time would therefore be considered slow. There are algorithms which take time slower than polynomial time ("super-polynomial time") but faster than exponential time ("sub-exponential time"). These are also considered "slow". One example is the best known algorithm for integer factorization. See also: Computational complexity theory Polynomial time Algorithm Big O notation.
Big O notation - depend on the precise details of the implementation and the hardware it runs on, so they should also be neglected. Big O notation captures what remains: we write T(n) = O(n2) and say that the algorithm has order of n2 time complexity. Properties If a function f(n) can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example . In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. O(nc) and O(cn) are very different. The latter grows much, much faster, no matter how big the constant c is. A function that grows faster than any power of n is called superpolynomial. One.
