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Fermat's Little Theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If is not divisible by , that is if is coprime to , 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 = 2 and = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. 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 ... [...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] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, ''n'', check whether it is evenly divisible by any prime number between 2 and (i.e. that the division leaves no remainder). If so, then ''n'' is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For example, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Primality Test
In computational number theory, the Lucas test is a primality test for a natural number ''n''; it requires that the prime factors of ''n'' − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that ''n'' is prime. Concepts Let ''n'' be a positive integer. If there exists an integer ''a'', 1 < ''a'' < ''n'', such that : and for every prime factor ''q'' of ''n'' − 1 : then ''n'' is prime. If no such number ''a'' exists, then ''n'' is either 1, 2, or . The reason for the correctness of this claim is as follows: if the first equivalence holds for ''a'', we can deduce that ''a'' and ''n'' are |
Carmichael Number
In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers b which are relatively prime to n. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 ( Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short). They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. Overview Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''b'', the number ''b'' − ''b'' is an integer multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Converse
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the converse of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange's Theorem (group Theory)
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group , the order (number of elements) of every subgroup of divides the order of . 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 :H/math>, defined as the number of left cosets of H in G. This variant holds even if G is infinite, provided that , G, , , H, , and :H/math> 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 left cosets form a partition of . Each left coset has the same cardinality as because x \mapsto ax defines a bijection H \to aH (the inverse is y \mapsto a^y). The number of left cosets is the index . By the previous three sentences, :\left, G\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael's Theorem
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind ''U''''n''(''P'', ''Q'') with relatively prime parameters ''P'', ''Q'' and positive discriminant, an element ''U''''n'' with ''n'' ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = ''U''12(1, −1) = 144 and its equivalent ''U''12(−1, −1) = −144. In particular, for ''n'' greater than 12, the ''n''th Fibonacci number F(''n'') has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof. Statement Given two relatively prime integers ''P'' and ''Q'', such that D=P^2-4Q>0 and , let be the Lucas sequence of the first kind defined by :\begin U_0(P,Q)&=0, \ ... [...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 the multiplicative group of integers modulo . The Carmichael function is named after the American mathematician 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 following table compares the first 36 values of with Euler's totient function (in bold if they are different; the s such that they are different are listed in ). Numerical examples # Carmichael's function at 5 is 4, , because for any number 0 |
Modular 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 are 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] |
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'' modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA Cryptosystem
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ) (the British signals intelligence agency) by the English mathematician Clifford Cocks. That system was declassified in 1997. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers. The security of RSA relies on the practical difficulty of factoring the product of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
