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Fermat's Factorization Method
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: :N = a^2 - b^2. That difference is algebraically factorable as (a+b)(a-b); if neither factor equals one, it is a proper factorization of ''N''. Each odd number has such a representation. Indeed, if N=cd is a factorization of ''N'', then :N = \left(\frac\right)^2 - \left(\frac\right)^2 Since ''N'' is odd, then ''c'' and ''d'' are also odd, so those halves are integers. (A multiple of four is also a difference of squares: let ''c'' and ''d'' be even.) In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either. Basic method One tries various values of ''a'', hoping that a^2-N = b^2, a square. FermatFactor(N): ''// N should be odd'' a ← b2 ← a*a - N repeat until b2 is a square: a ← a + 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Domin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization Of Polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Gaussian Integer Factorizations
A Gaussian integer is either the zero, one of the four units (±1, ±''i''), a Gaussian prime or composite. The article is a table of Gaussian Integers followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes. Note that there are rational primes which are not Gaussian primes. A simple example is the rational prime 5, which is factored as in the table, and therefore not a Gaussian prime. Conventions The second column of the table contains only integers in the first quadrant, which means the real part ''x'' is positive and the imaginary part ''y'' is non-negative. The table might have been further reduced to the integers in the first octant of the complex plane using the symmetry . The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry , fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Program Synthesis
In computer science, program synthesis is the task to construct a program that provably satisfies a given high-level formal specification. In contrast to program verification, the program is to be constructed rather than given; however, both fields make use of formal proof techniques, and both comprise approaches of different degrees of automatization. In contrast to automatic programming techniques, specifications in program synthesis are usually non-algorithmic statements in an appropriate logical calculus. Origin During the Summer Institute of Symbolic Logic at Cornell University in 1957, Alonzo Church defined the problem to synthesize a circuit from mathematical requirements. Even though the work only refers to circuits and not programs, the work is considered to be one of the earliest descriptions of program synthesis and some researchers refer to program synthesis as "Church's Problem". In the 1960s, a similar idea for an "automatic programmer" was explored by researchers in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Factorization Method
Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number 1000009 can be written as 1000^2 + 3^2 or as 972^2 + 235^2 and Euler's method gives the factorization 1000009 = 293 \cdot 3413. The idea that two distinct representations of an odd positive integer may lead to a factorization was apparently first proposed by Marin Mersenne. However, it was not put to use extensively until one hundred years later by Euler. His most celebrated use of the method that now bears his name was to factor the number 1000009, which apparently was previously thought to be prime even though it is not a pseudoprime by any major primality test. Euler's factorization method is more effective than Fermat's for integers whose factors are not close together and potentially much more efficient than trial division if one can find representations of numbers as sums of two squares reasonably easily. Euler's development ul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial . Factorization is not usually considered meaningful within number systems possessing division ring, division, such as the real number, real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by Greek mathematics, ancient Greek mathematicians in the case of integers. They proved the fundamental theorem o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Factorisation
In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorisation. The Schützenberger theorem relates the definition in terms of a multiplicative property to an additive property. Let ''A''* be the free monoid on an alphabet ''A''. Let ''X''''i'' be a sequence of subsets of ''A''* indexed by a totally ordered index set ''I''. A factorisation of a word ''w'' in ''A''* is an expression :w = x_ x_ \cdots x_ \ with x_ \in X_ and i_1 \ge i_2 \ge \ldots \ge i_n. Some authors reverse the order of the inequalities. Chen–Fox–Lyndon theorem A Lyndon word over a totally ordered alphabet ''A'' is a word that is lexicographically less than all its rotations.Lothaire (1997) p.64 The Chen–Fox–Lyndon theorem states that every string may be formed in a unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FOIL Rule
In secondary school, ''FOIL'' is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The word ''FOIL'' is an acronym for the four terms of the product: * First ("first" terms of each binomial are multiplied together) * Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second) * Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second) * Last ("last" terms of each binomial are multiplied) The general form is : (a + b)(c + d) = \underbrace_\text + \underbrace_\text + \underbrace_\text + \underbrace_\text. Note that is both a "first" term and an "outer" term; is both a "last" and "inner" term, and so forth. The order of the four terms in the sum is not important and need not match the order of the letters in the word FOIL. History The FOIL method is a special case of a more general method for multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Theorem
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial f(x) has a factor (x - \alpha) if and only if f(\alpha)=0 (i.e. \alpha is a root). Factorization of polynomials Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent. The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:. # Deduce the candidate of zero a of the polynomial f from its leading coefficient a_n and constant term a_0. (See Rational Root Theorem.) # Use the factor theorem to conclude that (x-a) is a factor of f(x). # Compute the polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
