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Integers Modulo N
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 can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clock Group
A clock or chronometer is a device that measures and displays time. The clock is one of the oldest Invention, human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month, and the year. Devices operating on several physical processes have been used over the Millennium, millennia. Some predecessors to the modern clock may be considered "clocks" that are based on movement in nature: A sundial shows the time by displaying the position of a shadow on a flat surface. There is a range of duration timers, a well-known example being the hourglass. Water clocks, along with sundials, are possibly the oldest time-measuring instruments. A major advance occurred with the invention of the verge escapement, which made possible the first mechanical clocks around 1300 in Europe, which kept time with oscillating timekeepers like balance wheels., pp. 103–104., p. 31. Traditionally, in horology (the study of timekeeping), the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (ring Theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Residue
In number theory, an integer ''q'' is a quadratic residue modulo operation, modulo ''n'' if it is Congruence relation, congruent to a Square number, perfect square modulo ''n''; that is, if there exists an integer ''x'' such that :x^2\equiv q \pmod. Otherwise, ''q'' is a quadratic nonresidue modulo ''n''. Quadratic residues are used in applications ranging from acoustical engineering to cryptography and the Integer factorization, factoring of large numbers. History, conventions, and elementary facts Fermat, Euler, Joseph Louis Lagrange, Lagrange, Adrien-Marie Legendre, 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 says that if the context makes it clear, the adjective "quadratic" may be dropped. For a giv ... [...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] |
Lagrange's Theorem (number Theory)
In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime ''p''. More precisely, it states that for all integer polynomials \textstyle f \in \mathbb /math>, either: * every coefficient of is divisible by ''p'', or * p \mid f(x) has at most solutions in , where is the degree of . This can be stated with congruence classes as follows: for all polynomials \textstyle f \in (\mathbb/p\mathbb) /math> with ''p'' prime, either: * every coefficient of is null, or * f(x)=0 has at most solutions in \mathbb/p\mathbb . If ''p'' is not prime, then there can potentially be more than solutions. Consider for example ''p=8'' and the polynomial ''f(x)=x'−1'', where ''1, 3, 5, 7'' are all solutions. Proof Let \textstyle f \in \mathbb /math> be an integer polynomial, and write the polynomial obtained by taking its coefficients . Then, for all integers ''x'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chinese Remainder Theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is sometimes called Sunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared in '' Sunzi Suanjing'', a Chinese manuscript written during the 3rd to 5th century CE. This first statement was restricted to the following example: If one knows that the remainder of ''n'' divided by 3 is 2, the remainder of ''n'' divided by 5 is 3, and the remainder of ''n'' divided by 7 is 2, then with no other information, one can determine the remainder of ''n'' divided by 105 (the product of 3, 5, and 7) without knowing the value of ''n''. In this example, the remainder is 23. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson's Theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of modular arithmetic), the factorial (n - 1)! = 1 \times 2 \times 3 \times \cdots \times (n - 1) satisfies :(n-1)!\ \equiv\; -1 \pmod n exactly when ''n'' is a prime number. In other words, any integer ''n'' > 1 is a prime number if, and only if, (''n'' − 1)! + 1 is divisible by ''n''. History The theorem was first stated by Ibn al-Haytham . Edward Waring announced the theorem in 1770 without proving it, crediting his student John Wilson for the discovery. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but never published it. Example For each of the values of ''n'' from 2 to 30, the following table shows the number (''n'' −&thinsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, then a^ is congruent to 1 modulo , where \varphi denotes Euler's totient function; that is :a^ \equiv 1 \pmod. In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where is not prime. The converse of Euler's theorem is also true: if the above congruence is true, then a and n must be coprime. The theorem is further generalized by some of Carmichael's theorems. The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7^, i.e. 7^ \pmod. The integers 7 and 10 are coprime, and \varphi( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Little Theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , then , and is an integer multiple of . If is not divisible by , that is, if is coprime to , then Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: a^ \equiv 1 \pmod p. For example, if and , then , and is a multiple of . Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Euclidean Algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's identity, which are integers ''x'' and ''y'' such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are coprime. With that provision, ''x'' is the modular multiplicative inverse of ''a'' modulo ''b'', and ''y'' is the modular multiplicative inverse of ''b'' mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout's Identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that and ; equality occurs only if one of and is a multiple of the other. As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as , with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Multiplicative Inverse
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congruence is written as :ax \equiv 1 \pmod, which is the shorthand way of writing the statement that divides (evenly) the quantity , or, put another way, the remainder after dividing by the integer is 1. If does have an inverse modulo , then there is an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to (i.e., in 's congruence class) has any element of 's congruence class as a modular multiplicative inverse. Using the notation of \overline to indicate the congruence class containing , this can be expressed by saying that the ''modulo multiplicative inverse'' of the congruence class \overline is the congruence class \overline such that: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
