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Shanks's Square Forms Factorization
Shanks's square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends on finding integers x and y such that x^2-y^2=N, where N is the integer to be factored. An improvement (noticed by Kraitchik) is to look for integers x and y such that x^2\equiv y^2\pmod. Finding a suitable pair (x, y) does not guarantee a factorization of N, but it implies that N is a factor of x^2-y^2=(x-y)(x+y), and there is a good chance that the prime divisors of N are distributed between these two factors, so that calculation of the greatest common divisor of N and x-y will give a non-trivial factor of N. A practical algorithm for finding pairs (x,y) which satisfy x^2\equiv y^2\pmod was developed by Shanks, who named it Square Forms Factorization or SQUFOF. The algorithm can be expressed in terms of continued fractions or in terms of quadratic forms. Although there are now much more effi ... [...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] |
Daniel Shanks
Daniel Shanks (January 17, 1917 – September 6, 1996) was an American mathematician who worked primarily in numerical analysis and number theory. He was the first person to compute π to 100,000 decimal places. Life and education Shanks was born on January 17, 1917, in Chicago, Illinois. He is not related to the English mathematician William Shanks, who was also known for his computation of π. He earned his Bachelor of Science degree in physics from the University of Chicago in 1937, and a Ph.D. in Mathematics from the University of Maryland in 1954. Prior to obtaining his PhD, Shanks worked at the Aberdeen Proving Ground and the Naval Ordnance Laboratory, first as a physicist and then as a mathematician. During this period he wrote his PhD thesis, which completed in 1949, despite having never taken any graduate math courses. After earning his PhD in mathematics, Shanks continued working at the Naval Ordnance Laboratory and the Naval Ship Research and Development Center at Da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Maurice Kraitchik
Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser. His main interests were the theory of numbers and recreational mathematics. He was born to a Jewish family in Minsk. He wrote several books on number theory during 1922–1930 and after the war, and from 1931 to 1939 edited ''Sphinx'', a periodical devoted to recreational mathematics. During World War II, he emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." Kraïtchik was ''agrégé'' of the Free University of Brussels, engineer at the Société Financière de Transports et d'Entreprises Industrielles (Sofina), and director of the Institut des Hautes Etudes de Belgique. He died in Brussels. Kraïtchik is famous for having inspired the two envelopes problem The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Divisor
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] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Václav Šimerka
Václav Šimerka (20 December 1819 – 26 December 1887) was a Bohemian mathematician, priest, physicist, and philosopher. He wrote the first Czech text on calculus and is credited for discovering the first seven Carmichael numbers, from 561 to 8911, in 1885. Biography Šimerka was born on 20 December 1819 in Vysoké Veselí in Bohemia to a family of coopers of businessman Petr Šimerka and his wife Terezie. After attending school in Jičín, he studied in the University of Prague's Faculty of Philosophy from 1839 to 1841. There, he studied mathematics under Jakob Philipp Kulik and astronomy under and practical geometry under Adam Bittner and also obligatory teaching of religion, philosophy, mathematics, Latin philology, natural science, physics, moral philosophy and history. After graduating in Prague, Šimerka studied in the Theological Seminary in Hradec Králové. Šimerka was ordained on 25 July 1845 and then became a chaplain in Žlunice near Jičín. He only spent a short ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
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 pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
