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Multiplicative Order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order of ''a'' modulo ''n'' is the order of ''a'' in the multiplicative group of the units in the ring of the integers modulo ''n''. The order of ''a'' modulo ''n'' is sometimes written as \operatorname_n(a). Example The powers of 4 modulo 7 are as follows: : \begin 4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\ 4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\ 4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\ 4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\ 4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\ 4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\ \vdots\end The smallest positive integer ''k'' such that 4''k'' ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange's Theorem (group Theory)
In the mathematics, mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group , then , H, is a divisor of , G, , i.e. the order of a group, order (number of elements) of every subgroup H divides the order of group G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup H of a finite group G, not only is , G, /, H, an integer, but its value is the index of a subgroup, index [G:H], defined as the number of left cosets of H in G. This variant holds even if G is infinite, provided that , G, , , H, , and [G:H] are interpreted as cardinal numbers. Proof The left cosets of in are the equivalence classes of a certain equivalence relation on : specifically, call and in equivalent if there exists in such that . Therefore, the set of left cosets forms a Partition of a set, partition of . Each left coset has the same cardinality as because x \mapsto ax defines a bijection H \to aH ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm
In mathematics, for given real numbers a and b, the logarithm \log_b(a) is a number x such that b^x=a. Analogously, in any group G, powers b^k can be defined for all integers k, and the discrete logarithm \log_b(a) is an integer k such that b^k=a. In arithmetic modulo an integer m, the more commonly used term is index: One can write k=\mathbb_b a \pmod (read "the index of a to the base b modulo m") for b^k \equiv a \pmod if b is a primitive root of m and \gcd(a,m)=1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. In cryptography, the computational complexity of the discrete logarithm problem, along with its application, was first proposed in the Diffie–Hellman problem. Several important algorithms in public-key cryptography, such as ElGamal, base their security on the hardness assumption that the discrete logarithm problem (DLP) over carefully chosen groups has no efficient solution. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosetta Code
Rosetta Code is a wiki-based programming chrestomathy website with implementations of common algorithms and solutions to various computer programming, programming problems in many different programming languages. It is named for the Rosetta Stone, which has the same text inscribed on it in three languages, and thus allowed Egyptian hieroglyphs to be deciphered for the first time. Website Rosetta Code was created in 2007 by Michael Mol. The site's content is licensed under the GNU Free Documentation License 1.2, though some components may be dual-licensed under more permissive terms. The Rosetta Code web repository illustrates how desired functionality is implemented very differently in various Programming_paradigm, programming paradigms, and how "the same" task is accomplished in different programming languages. , Rosetta Code has: * 1,266 computer programming tasks (or problems) * 404 additional draft programming tasks * 933 computer programming languages that are used to solve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Language
The Wolfram Language ( ) is a proprietary, very high-level multi-paradigm programming language developed by Wolfram Research. It emphasizes symbolic computation, functional programming, and rule-based programming and can employ arbitrary structures and data. It is the programming language of the mathematical symbolic computation program Mathematica. History The Wolfram Language was part of the initial version of Mathematica in 1988. Symbolic aspects of the engine make it a computer algebra system. The language can perform integration, differentiation, matrix manipulations, and solve differential equations using a set of rules. Also, the initial version introduced the notebook model and the ability to embed sound and images, according to Theodore Gray's patent. Wolfram also added features for more complex tasks, such as 3D modeling. A name was finally adopted for the language in 2013, as Wolfram Research decided to make a version of the language engine free for Raspberry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxima CAS
Maxima () is a software package for performing computer algebra calculations in mathematics and the physical sciences. It is written in Common Lisp and runs on all POSIX platforms such as macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. It is free software released under the terms of the GNU General Public License (GPL). History Maxima is based on a 1982 version of Macsyma, which was developed at MIT with funding from the United States Department of Energy and other government agencies. A version of Macsyma was maintained by Bill Schelter from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release his version under the GPL. That version, now called Maxima, is maintained by an independent group of users and developers. Maxima does not include any of the many modifications and enhancements made to the commercial version of Macsyma during 1982–1999. Though the core functionality remains similar, code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael Function
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that :a^m \equiv 1 \pmod holds for every integer coprime to . In algebraic terms, is the exponent of a group, exponent of the multiplicative group of integers modulo n, multiplicative group of integers modulo . As this is a Abelian group#Finite abelian groups, finite abelian group, there must exist an element whose Cyclic group#Definition and notation, order equals the exponent, . Such an element is called a primitive -root modulo . The Carmichael function is named after the American mathematician Robert Daniel Carmichael, Robert Carmichael who defined it in 1910. It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo is , where is Euler's totient function. Since the order of an element of a finite group divides the order of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, generated by a single element. That is, it is a set (mathematics), set of Inverse element, invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer Exponentiation, power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a ''Generating set of a group, generator'' of the group. Every infinite cyclic group is isomorphic to the additive group \Z, the integers. Every finite cyclic group of Order (group theory), order n is isomorphic to the additive group of Quotient group, Z/''n''Z, the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Root Modulo N
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo'' if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the '' Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. A primitive root exists if and only if ''n'' is 1, 2, 4, ''p''''k'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
