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Blum Integer
In mathematics, a natural number ''n'' is a Blum integer if is a semiprime for which ''p'' and ''q'' are distinct prime numbers congruent to 3 mod 4.Joe Hurd, Blum Integers (1997), retrieved 17 Jan, 2011 from http://www.gilith.com/research/talks/cambridge1997.pdf That is, ''p'' and ''q'' must be of the form , for some integer ''t''. Integers of this form are referred to as Blum primes. Goldwasser, S. and Bellare, M.br>"Lecture Notes on Cryptography". Summer course on cryptography, MIT, 1996-2001 This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are : 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ... The integers were named for computer scientist Manuel Blum. Properties Given a Blum integer, ''Q''''n'' the set of all quadratic residues modulo ''n'' and coprime to ''n'' and . Then: *''a'' has f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
93 (number)
93 (ninety-three) is the natural number following 92 and preceding 94. In mathematics 93 is: * the twenty-eighth distinct semiprime and the ninth of the form (3.q). * the first number in the third triplet of consecutive semiprimes, 93, 94, and 95. * a Blum integer, since its two prime factors, 3 and 31 are both Gaussian primes. * a repdigit in base 5 (3335), and 30 (3330). * palindromic in bases 2, 5, and 30. * a lucky number. * a cake number. * an idoneal number. There are 93 different cyclic Gilbreath permutations on 11 elements, and therefore there are 93 different real periodic points of order 11 on the Mandelbrot set. In other fields Ninety-three is: *The atomic number of neptunium, an actinide. * The code for international direct dial phone calls to Afghanistan. * One of two ISBN Group Identifiers for books published in India. * The number of the French department Seine-Saint-Denis, and as such used by many French gangsta rappers and those emulating their speech. In cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( \left(\sqrt + o(1)\right)(\ln n)^(\ln \ln n)^\right) =L_n\left .html"_;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in_L-notation.html" ;"title="">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation">">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation), where is the natural logarithm. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MPQS
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by Carl Pomerance in 1981 as an improvement to Schroeppel's linear sieve. Basic aim The algorithm attempts to set up a congruence of squares modulo ''n'' (the integer to be factorized), which often leads to a factorization of ''n''. The algorithm works in two phases: the ''data collection'' phase, where it collects information that may lead to a congruence of squares; and the ''data processing'' phase, where it puts all the data it has collected into a matrix and solves it to obtain a congruence of squares ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of −1 is a quadratic residue, and if ''k'' is a quadratic residue modulo a coprime ''n'', then , but not all entries with a Jacobi symbol of 1 (see the and rows) are quadratic residues. Notice also that when either ''n'' or ''k'' is a square, all values are nonnegative. The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography. Definition For any integer ''a'' and any positive odd integer ''n'', the Jacobi symbol is defined as the product of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic nonresidue modulo ''n''. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers. History, conventions, and elementary facts Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's ''Disquisitiones Arithmeticae'' (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuel Blum
Manuel Blum (born 26 April 1938) is a Venezuelan-American computer scientist who received the Turing Award in 1995 "In recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking". Education Blum was born to a Jewish family in Venezuela. Blum was educated at MIT, where he received his bachelor's degree and his master's degree in electrical engineering in 1959 and 1961 respectively, and his Ph.D. in mathematics in 1964 supervised by Marvin Minsky.. Career Blum worked as a professor of computer science at the University of California, Berkeley until 2001. From 2001 to 2018, he was the Bruce Nelson Professor of Computer Science at Carnegie Mellon University, where his wife, Lenore Blum, was also a professor of Computer Science. In 2002, he was elected to the United States National Academy of Sciences. In 2006, he was elected a member of the National Academy of Engineering for contributions to abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
177 (number)
177 (one hundred ndseventy-seven) is the natural number following 176 and preceding 178. In mathematics It is a Leyland number since . It is a 60-gonal number, and an arithmetic number, since the mean of its divisors ( 1, 3, 59 and 177) is equal to 60, an integer. 177 is a Leonardo number, part of a sequence of numbers closely related to the Fibonacci numbers. In graph enumeration, there are 177 undirected graphs (not necessarily connected) that have seven edges and no isolated vertices, and 177 rooted trees with ten nodes and height at most three. There are 177 ways of re-connecting the (labeled) vertices of a regular octagon into a star polygon that does not use any of the octagon edges. In other fields 177 is the second highest score for a flight of three darts Darts or dart-throwing is a competitive sport in which two or more players bare-handedly throw small sharp-pointed missiles known as darts at a round target known as a dartboard. Points can be scored by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
161 (number)
161 (one hundred ndsixty-one) is the natural number following 160 and preceding 162. In mathematics * 161 is the sum of five consecutive prime numbers: 23, 29, 31, 37, and 41 * 161 is a hexagonal pyramidal number. * 161 is a semiprime. Since its prime factors 7 and 23 are both Gaussian primes, 161 is a Blum integer. * 161 is a palindromic number * is a commonly used rational approximation of the square root of 5 and is the closest fraction with denominator 88 (88 is code for Heil Hitler among neo-nazis, as H=8) See also * Anti-Fascist Action * List of highways numbered 161 * United Nations Security Council Resolution 161 United Nations Security Council Resolution 161 was adopted on February 21, 1961. After noting the killings of Patrice Lumumba, Maurice Mpolo and Joseph Okito and a report of the United Nations Secretary-General, Secretary-General's Special Represe ... * United States Supreme Court cases, Volume 161 External links Number Facts and Trivia: 161The Numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
141 (number)
141 (one hundred ndforty-one) is the natural number following 140 and preceding 142. In mathematics 141 is: *a centered pentagonal number. *the sum of the sums of the divisors of the first 13 positive integers. *the second ''n'' to give a prime Cullen number (of the form ''n''2''n'' + 1). *an undulating number in base 10, with the previous being 131, and the next being 151. *the sixth hendeca gonal (11-gonal) number. *a semiprime: a product of two prime numbers, namely 3 and 47. Since those prime factors are Gaussian primes, this means that 141 is a Blum integer. * a Hilbert prime In the military * The Lockheed C-141 Starlifter was a United States Air Force military strategic airlifter * K-141 ''Kursk'' was a Russian nuclear cruise missile submarine, which sank in the Barents Sea on 12 August 2000 * was a United States Navy ship during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
133 (number)
133 (one hundred ndthirty-three) is the natural number following 132 and preceding 134. In mathematics 133 is an ''n'' whose divisors (excluding ''n'' itself) added up divide φ(''n''). It is an octagonal number and a happy number. 133 is a repdigit in base 11 (111) and base 18 (77), whilst in base 20 it is a cyclic number formed from the reciprocal of the number three. 133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer. 133 is the number of compositions of 13 into distinct parts. In the military * Douglas C-133 Cargomaster was a United States cargo aircraft built between 1956 and 1961 * is a heavy landing craft which launched in 1972 * was a United States Navy ''Mission Buenaventura''-class fleet oilers during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy ''General G. O. Squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
129 (number)
129 (one hundred ndtwenty-nine) is the natural number following 128 and preceding 130. In mathematics 129 is the sum of the first ten prime numbers. It is the smallest number that can be expressed as a sum of three squares in four different ways: 11^2+2^2+2^2, 10^2+5^2+2^2, 8^2+8^2+1^2, and 8^2+7^2+4^2. 129 is the product of only two primes, 3 and 43, making 129 a semiprime. Since 3 and 43 are both Gaussian primes, this means that 129 is a Blum integer. 129 is a repdigit in base 6 (333). 129 is a happy number. 129 is a centered octahedral number. In the military * Raytheon AGM-129 ACM (Advanced Cruise Missile) was a low observable, sub-sonic, jet-powered, air-launched cruise missile used by the United States Air Force * Soviet submarine K-129 (1960) was a Soviet Pacific Fleet nuclear submarine that sank in 1968 * was a United States Navy ''Mission Buenaventura''-class fleet oilers during World War II * was a ''Crosley-class high speed transport of the United States Navy * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
