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Proth's Theorem
In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers (sometimes called Proth Numbers of the First Kind). For ''Proth Numbers of the Second Kind'', see Riesel numbers. It states that for any integer ''p'' that is a Proth number (of the first kind) - an integer of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'' - and if there exists an integer ''a'' for which Euler's criterion is ''-1'', thus: : then ''p'' is . In this case, ''p'' is called a Proth prime. The is also true: if ''p'' is composite then no such ''a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
False Negative
A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result incorrectly indicates the absence of a condition when it is actually present. These are the two kinds of errors in a binary test, in contrast to the two kinds of correct result (a and a ). They are also known in medicine as a false positive (or false negative) diagnosis, and in statistical classification as a false positive (or false negative) error. In statistical hypothesis testing, the analogous concepts are known as type I and type II errors, where a positive result corresponds to rejecting the null hypothesis, and a negative result corresponds to not rejecting the null hypothesis. The terms are often used interchangeably, but there are differences in detail and interpretation due to the differences between medical testing and sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Common tasks such as 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 started in January 1996. It was followed in 1997 by distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Prime Pages
The PrimePages is a website about prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...s originally created by Chris Caldwell at the University of Tennessee at Martin who maintained it from 1994 to 2023. 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 PrimePages has articles on primes and primality testing. 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 1984. On March 11, 2023, the PrimePages moved from primes.utm.edu to t5k.or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
193 (number)
193 (one hundred ndninety-three) is the natural number following 192 and preceding 194. In mathematics 193 is the number of compositions of 14 into distinct parts. In decimal, it is the seventeenth full repetend prime, or ''long prime''. * It is the only odd prime p known for which 2 is not a primitive root of 4p^2 + 1. * It is the thirteenth Pierpont prime, which implies that a regular 193-gon can be constructed using a compass, straightedge, and angle trisector. * It is part of the fourteenth pair of twin primes (191, 193), the seventh trio of prime triplets (193, 197, 199), and the fourth set of prime quadruplets (191, 193, 197, 199). Aside from itself, the '' friendly giant'' (the largest sporadic group) holds a total of 193 conjugacy classes. It also holds at least 44 maximal subgroups aside from the double cover of \mathbb (the forty-fourth prime number is 193). 193 is also the eighth numerator of convergents to Euler's number; correct to three decimal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
113 (number)
113 (one hundred ndthirteen) is the natural number following 112 and preceding 114. Mathematics * 113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 7\times 2^+1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In decimal, this prime is a primeval number and a permutable prime with 131 and 311. *113 is a highly cototient number and a centered square number. *113 is the denominator of 355/113, an accurate approximation to . Other uses *113 is also the atomic number of nihonium. * A113 is a Pixar recurring inside joke or Easter Egg, e.g.: (WALL-E ''WALL-E'' (stylized with an interpunct as ''WALL·E'') is a 2008 American animated Romance film, romantic science fiction film produced by Pixar Animation Studios for Walt Disney Pictures. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
97 (number)
97 (ninety-seven) is the natural number following 96 (number), 96 and preceding 98 (number), 98. It is a prime number and the only prime in the nineties. In mathematics 97 is: * the 25th prime number (the largest two-digit prime number in Base 10, base 10), following 89 (number), 89 and preceding 101 (number), 101. * a Proth prime and a Pierpont prime as it is 3 × 25 + 1. * the eleventh member of the Mian–Chowla sequence. * a self number in base 10, since there is no integer that added to its own digits, adds up to 97. * the smallest odd prime that is not a cluster prime. * the highest two-digit number where the sum of its digits is a square. * the number of primes less than 29. * The numbers 97, 907, 9007, 90007 and 900007 are all primes, and they are all happy primes. However, 9000007 (read as ''nine million seven'') is composite number, composite and has the factorization 277 (number), 277 × 32491. * an emirp with 79 (number), 79. * an isolated p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
41 (number)
41 (forty-one) is the natural number following 40 (number), 40 and preceding 42 (number), 42. In mathematics 41 is: * the 13th smallest prime number. The next is 43 (number), 43, making both twin primes. * the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13). * a regular prime * a Ramanujan prime * a harmonic prime * a good prime * the 12th Supersingular prime (moonshine theory), supersingular prime * a Newman–Shanks–Williams prime. * the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, . * an Eisenstein prime, with no imaginary part and real part of the form 3''n'' − 1. * a Proth prime as it is 5 × 23 + 1. * the smallest Lucky numbers of Euler, lucky number of Euler: the polynomial yields primes for all the integers ''k'' with . * the sum of two Square number, squares (42 + 52), which makes it a centered square number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). 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, etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see ''Polynomial greatest common divisor'') and other commutative rings (see ' below). Overview Definition The ''greatest common divisor'' (GCD) of integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Solovay–Strassen Primality Test
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic primality test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA cryptosystem. Concepts Euler proved that for any odd prime number ''p'' and any integer ''a'', :a^ \equiv \left(\frac\right) \pmod p where \left(\tfrac\right) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to \left(\tfrac\right), where ''n'' can be any odd integer. The Jacobi symbol can be computed in time O((log ''n'')²) using Jacobi's generalization of the law of quadratic reciprocity. Given an odd number ''n'' one can contemplate whether or not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fermat Number
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a natural number, positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3 (number), 3, 5 (number), 5, 17 (number), 17, 257 (number), 257, 65537 (number), 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... . If 2''k'' + 1 is Prime number, prime and , then ''k'' itself must be a power of 2, so is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are , , , , and . Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for . Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
