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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 ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'', and if there exists an ''a'' for which : then ''p'' is . 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. In practice, however, a quadratic nonresidue of ''p'' is found via a [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
193 (number)
193 (one hundred ndninety-three) is the natural number following 192 and preceding 194. In mathematics 193 is a prime number, and a Pierpont prime, implying that a 193-gon can be constructed using a compass, straightedge, and angle trisector. It is the only odd prime p known for which 2 is not a primitive root of 4p^2 + 1. It is the number of compositions of 14 into distinct parts. It is also part of the prime quadruplet (191, 193, 197, 199). See also * 193 (other) 193 A.D. is a year. 193 may also refer to: * 193 Ambrosia * Connecticut Route 193 * Maryland Route 193 * West Virginia Route 193 * Alabama State Route 193 * California State Route 193 * Ohio State Route 193 * Georgia State Route 193 * Maine S ... References {{DEFAULTSORT:193 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems About Prime Numbers
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Tests
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
François Proth
François Proth (22 March 1852 – 21 January 1879) was a French self-taught mathematician farmer who lived in Vaux-devant-Damloup near Verdun, France. He stated four primality-related theorems. The most famous of these, Proth's theorem, can be used to test whether a Proth number (a number of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François ...s; they continue to be of importance in the computational search for large prime numbers. Proth also formulated Gilbreath's conjecture on successive differences of primes, 80 years prior to Gilbreath, but his proof of the conjecture turned out to be erroneous.. The cause of Proth's death is not known. Publications *. *. *. *. References 1852 births 1879 deaths 19th-century French mathematicians {{france-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pépin's Test
In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime 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 .... It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin. Description of the test Let F_n=2^+1 be the ''n''th Fermat number. Pépin's test states that for ''n'' > 0, :F_n is prime if and only if 3^\equiv-1\pmod. The expression 3^ can be evaluated modulo F_n by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat numbers can be tested in a reasonable amount of time and space. Other bases may be used in place of 3. These bases are: :3, 5, 6, 7, 10, 12, 14, 20, 24, 27, 28, 39, 40, 41, 45, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pocklington Primality Test
In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N - 1 to prove that an integer N is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of N - 1. Pocklington criterion The basic version of the test relies on the Pocklington theorem (or Pocklington criterion) which is formulated as follows: Let N > 1 be an integer, and suppose there exist natural numbers and such that Then is prime. Note: Equation () is simply a Fermat primality test. If we find ''any'' value of , not divisible by , such that equation () is false, we may immediately conclude that is not prime. (This divisibility condition is not explicitly stated because it is implied by equation ().) For example, let N = 35. With a = 2, we find that a^ \equiv 9 \pmod. This is enough to prove t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volunteer Computing
Volunteer computing is a type of distributed computing in which people donate their computers' unused resources to a research-oriented project, and sometimes in exchange for credit points. The fundamental idea behind it is that a modern desktop computer is sufficiently powerful to perform billions of operations a second, but for most users only between 10-15% of its capacity is used. Typical uses like basic word processing or web browsing leave the computer mostly idle. The practice of volunteer computing, which dates back to the mid-1990s, can potentially make substantial processing power available to researchers at minimal cost. Typically, a program running on a volunteer's computer periodically contacts a research application to request jobs and report results. A middleware system usually serves as an intermediary. History The first volunteer computing project was the Great Internet Mersenne Prime Search, which was started in January 1996. It was followed in 1997 by distribute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Pages
The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. , the 5,000th prime has around 412,000 digits.. Retrieved on 2018-02-12. The PrimePages has articles on primes and 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 whet ...ing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of titanic primes (primes with at least 1000 decimal digits) by Samuel Yates. In subsequent years, the whole top-5,000 has consisted o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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