Wilson's theoremIn 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 satisfies exactly when n is a prime number. In other words, any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n. This theorem was stated by Ibn al-Haytham (c. 1000 AD), and, in the 18th century, by the English mathematician John Wilson.
Greatest common divisorIn 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 . For example, the GCD of 8 and 12 is 4, that is, . 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.
CryptosystemIn cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term cipher (sometimes cypher) is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term cryptosystem is most often used when the key generation algorithm is important.
Arithmetic groupIn mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
Euclidean algorithmIn mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use.
Fermat's Last TheoremIn number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin.
GCD domainIn mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
Mathematics educationIn contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods.
Encryption softwareEncryption software is software that uses cryptography to prevent unauthorized access to digital information. Cryptography is used to protect digital information on computers as well as the digital information that is sent to other computers over the Internet. There are many software products which provide encryption. Software encryption uses a cipher to obscure the content into ciphertext. One way to classify this type of software is the type of cipher used.
Modular groupIn mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.
Fermat's little theoremFermat's little theorem states that if p is a prime number, then for any integer a, the number is an integer multiple of p. In the notation of modular arithmetic, this is expressed as For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is if a is coprime to p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.
Fermat numberIn mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
