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Double Mersenne Number
In mathematics, a double Mersenne number is a Mersenne prime, Mersenne number of the form :M_ = 2^-1 where ''p'' is prime number, prime. Examples The first four terms of the integer sequence, sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and Lists''at the Prime Pages. : :M_ = M_3 = 7 :M_ = M_7 = 127 :M_ = M_ = 2147483647 :M_ = M_ = 170141183460469231731687303715884105727 Double Mersenne primes A double Mersenne number that is prime number, prime is called a double Mersenne prime. Since a Mersenne number ''M''''p'' can be prime only if ''p'' is prime, (see Mersenne prime for a proof), a double Mersenne number M_ can be prime only if ''M''''p'' is itself a Mersenne prime. For the first values of ''p'' for which ''M''''p'' is prime, M_ is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of M_ have been found for ''p'' = 13, 17, 19, and mersenne prime 31. Thus, the smallest candidate for the next double Mersenne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Édouard Lucas
__NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lucas was born in Amiens and educated at the École Normale Supérieure. He worked in the Paris Observatory and later became a professor of mathematics at the Lycée Saint Louis and the Lycée Charlemagne in Paris. Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871. In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation :\sum_^ n^2 = M^2\; with ''N'' > 1 is when ''N'' = 24 and ''M'' = 70. This is known as the cannonball problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid out of them. It was not until 1918 that a proof (using elliptic functions) was found for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6 (number), 6, 28 (number), 28, 496 (number), 496 and 8128 (number), 8128. The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements, Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby \frac is an even perfect number whenever q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Double Exponential Function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b'' = 10: *''f''(x) = 1010x *''f''(0) = 10 *''f''(1) = 1010 *''f''(2) = 10100 = googol *''f''(3) = 101000 *''f''(100) = 1010100 = googolplex. Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(''x'')). The complex double exponential function is entire, because it is the composition of two entire functions f(x)=a^x=e^ and g(x)=b^x=e^. Double exponential sequences A sequence of positive integers (or real numbers) is said to have ''double exponential rate of growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cunningham Chain
In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes. Definition A Cunningham chain of the first kind of length ''n'' is a sequence of prime numbers (''p''1, ..., ''p''''n'') such that ''p''''i''+1 = 2''p''''i'' + 1 for all 1 ≤ ''i'' < ''n''. (Hence each term of such a chain except the last is a , and each term except the first is a ). It follows that : or, by setting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach Conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than but remains unproven despite considerable effort. History Origins On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (""), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to his seco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Beast With A Billion Backs
''Futurama: The Beast with a Billion Backs'' is a 2008 American direct-to-video adult animated science-fiction film based on the animated series ''Futurama'', and the second of four straight-to-DVD films that make up the show's fifth season. The film was released in the United States and Canada on June 24, 2008, followed by a UK release on June 30, 2008 and an Australian release on August 6, 2008. The title refers to a euphemism for sexual intercourse—" the beast with two backs". Comedy Central aired the film as a "four-part epic" on October 19, 2008. The movie won an Annie Award for "Best Animated Home Entertainment Production". Plot A month after the universe was ripped open, people decide to go on with their lives. Amy and Kif get married. Fry starts dating a girl named Colleen, but breaks up with her when he discovers she has many other boyfriends. At a conference, Professor Farnsworth proposes an expedition to investigate the anomaly. When Bender explores the anomal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Futurama
''Futurama'' is an American animated science fiction sitcom created by Matt Groening for the Fox Broadcasting Company and later revived by Comedy Central, and then Hulu. The series follows Philip J. Fry, who is cryogenically preserved for 1,000 years and revived on December 31, 2999. Fry finds work at the interplanetary delivery company Planet Express, working alongside the one-eyed mutant Leela and the robot Bender. The series was envisioned by Groening in the mid-1990s while working on ''The Simpsons''; he brought David X. Cohen aboard to develop storylines and characters to pitch the show to Fox. Following its initial cancellation by Fox, ''Futurama'' began airing reruns on Cartoon Network's Adult Swim programming block, which lasted from 2003 to 2007. It was revived in 2007 as four direct-to-video films, the last of which was released in early 2009. Comedy Central entered into an agreement with 20th Century Fox Television to syndicate the existing episodes and air ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Mersenne Conjecture
In mathematics, the Mersenne conjectures concern the characterization of a kind of prime numbers called Mersenne primes, meaning prime numbers that are a power of two minus one. Original Mersenne conjecture The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his ''Cogitata Physico-Mathematica'' (1644; see e.g. Dickson 1919) that the numbers 2^n - 1 were prime for ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 , and were composite for all other positive integers ''n'' ≤ 257. The first seven entries of his list (2^n - 1 for ''n'' = 2, 3, 5, 7, 13, 17, 19) had already been proven to be primes by trial division before Mersenne's time; only the last four entries were new claims by Mersenne. Due to the size of those last numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as the Lucas–Lehmer test, that Mers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime number, prime, or the Unit (ring theory), unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Number
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 should 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
