Concept
Prime number
Related concepts (96)
13 (number)
13 (thirteen) is the natural number following 12 and preceding 14. Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition. The number 13 is the sixth prime number.
AKS primality test
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". The algorithm was the first one which is able to determine in polynomial time, whether a given number is prime or composite and this without relying on mathematical conjectures such as the generalized Riemann hypothesis.
Hasse principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
Formula for primes
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. A simple formula is for positive integer , where is the floor function, which rounds down to the nearest integer. By Wilson's theorem, is prime if and only if . Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number .
11 (number)
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables. "Eleven" derives from the Old English ęndleofon, which is first attested in Bede's late 9th-century Ecclesiastical History of the English People. It has cognates in every Germanic language (for example, German elf), whose Proto-Germanic ancestor has been reconstructed as *ainalifa-, from the prefix *aina- (adjectival "one") and suffix *-lifa-, of uncertain meaning.
Proth's theorem
In number theory, Proth's theorem is a primality test for Proth numbers. It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which then p is prime. In this case p is called a Proth prime. This is a practical test because if p is prime, any chosen a has about a 50 percent chance of working, furthermore, since the calculation is mod p, only values of a smaller than p have to be taken into consideration.
Additive number theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers.
Linear congruential generator
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modular arithmetic by storage-bit truncation.
Mersenne Twister
The Mersenne Twister is a general-purpose pseudorandom number generator (PRNG) developed in 1997 by ja and Takuji Nishimura. Its name derives from the fact that its period length is chosen to be a Mersenne prime. The Mersenne Twister was designed specifically to rectify most of the flaws found in older PRNGs. The most commonly used version of the Mersenne Twister algorithm is based on the Mersenne prime . The standard implementation of that, MT19937, uses a 32-bit word length.
Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity.