1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Recursion
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Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence
3.
Futurama
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Futurama is an American animated science fiction sitcom created by Matt Groening for the Fox Broadcasting Company. The series was envisioned by Groening in the mid-1990s while working on The Simpsons, he later brought David X. Cohen aboard to develop storylines and characters to pitch the show to Fox. In the United States, the series aired on Fox from March 28,1999, to August 10,2003, Futurama also aired in reruns on Cartoon Networks Adult Swim from 2002 to 2007, until the networks contract expired. 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 the films as 16 new, half-hour episodes, constituting a fifth season. In June 2009, producing studio 20th Century Fox announced that Comedy Central had picked up the show for 26 new half-hour episodes, the show was renewed for a seventh season, with the first half airing in June 2012 and the second set for mid-2013. It was later revealed that the season would be the final season. The series finale aired on September 4,2013, while Groening has said he will try to get it picked up by another network, David X. Cohen stated that the episode Meanwhile would be the last episode of season 7 and also the series finale. Throughout its run, Futurama has received critical acclaim, the show has been nominated for 17 Annie Awards and 12 Emmy Awards, winning seven of the former and six of the latter. Futurama-related merchandise has also released, including a tie-in comic book series and video game, calendars. In 2013, TV Guide ranked Futurama as one of the top 60 Greatest TV Cartoons of All Time, Fox expressed a strong desire in the mid-1990s for Matt Groening to create a new series, and he began conceiving Futurama during this period. In 1996, he enlisted David X. Cohen, then a writer and producer for The Simpsons, the two spent time researching science fiction books, television shows, and films. When they pitched the series to Fox in April 1998, Groening and Cohen had composed many characters and story lines, Groening described trying to get the show on the air as by far the worst experience of my grown-up life. With The Simpsons the network has no input, Fox was particularly disturbed by the concept of suicide booths, Doctor Zoidberg, and Benders anti-social behavior. Groening explains, When they tried to give me notes on Futurama, I just said, and they said, Well, we dont do business that way anymore. And I said, Oh, well, thats the only way I do business, the episode I, Roommate was produced to address Foxs concerns, with the script written to their specifications. Fox strongly disliked the episode, but after negotiations, Groening received the same independence with Futurama, the name Futurama comes from a pavilion at the 1939 New York Worlds Fair. Designed by Norman Bel Geddes, the Futurama pavilion depicted how he imagined the world would look in 1959, many other titles were considered for the series, including Aloha, Mars. and Doomsville, which Groening notes were resoundly rejected, by everyone concerned with it. It takes approximately six to nine months to produce an episode of Futurama, the long production time results in several episodes being worked on simultaneously
4.
2,147,483,647
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The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 −1. It is one of four known double Mersenne primes. The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772, Euler used trial division, improving on Cataldis method, so that at most 372 divisions were needed. It thus improved upon the previous record-holding prime,6,700,417, also discovered by Euler, the number 2,147,483,647 remained the largest known prime until 1867. He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary, in fact a larger prime was discovered in 1855 by Thomas Clausen, though a proof was not provided. Furthermore,3,203,431,780,337 was proven to be prime in 1867, the number 2,147,483,647 is the maximum positive value for a 32-bit signed binary integer in computing. It is therefore the value for variables declared as integers in many programming languages. The appearance of the number often reflects an error, overflow condition, google later admitted that this was a joke. The data type time_t, used on operating systems such as Unix, is a signed integer counting the number of seconds since the start of the Unix epoch, and is often implemented as a 32-bit integer. The latest time that can be represented in this form is 03,14,07 UTC on Tuesday,19 January 2038 and this means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem. Also, this number is in most browsers the highest to accept positive or negative z-index in Cascading Style Sheets, power of two Prime curios,2147483647
5.
Goldbach's conjecture
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Goldbachs conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states, Every even integer greater than 2 can be expressed as the sum of two primes, the conjecture has been shown to hold up through 4 ×1018, but remains unproven despite considerable effort. A Goldbach number is an integer that can be expressed as the sum of two odd primes. The expression of an even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers,6 =3 +38 =3 +510 =3 +7 =5 +512 =7 +5. 100 =3 +97 =11 +89 =17 +83 =29 +71 =41 +59 =47 +53. He then proposed a second conjecture in the margin of his letter and he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time, a modern version of Goldbachs marginal conjecture is, Every integer greater than 5 can be written as the sum of three primes. In the letter dated 30 June 1742, Euler stated, Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, Goldbachs third version is the form in which the conjecture is usually expressed today. It is also known as the strong, even, or binary Goldbach conjecture, while the weak Goldbach conjecture appears to have been finally proved in 2013, the strong conjecture has remained unsolved. For small values of n, the strong Goldbach conjecture can be verified directly, for instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤105. With the advent of computers, many more values of n have been checked, one record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781. A very crude version of the heuristic argument is as follows. The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is an even integer and m is a number between 3 and n/2, then one might expect the probability of m and n − m simultaneously being prime to be 1 /. Since this quantity goes to infinity as n increases, we expect that every even integer has not just one representation as the sum of two primes, but in fact has very many such representations. This heuristic argument is somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd then n − m is odd, and if m is even, then n − m is even
6.
Futurama: The Beast with a Billion Backs
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Futurama, The Beast with a Billion Backs is a 2008 American animated science-fiction comedy film and the second of the four Futurama straight-to-DVD films. The film was released in the United States and Canada on June 24,2008, followed by a UK release on June 30,2008, 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, a month after the universe was ripped open in Futurama, Benders Big Score, people decide to go on with their lives. Fry starts dating a girl named Colleen, but breaks up with her when he discovers she has many more boyfriends, at a scientific conference, Professor Farnsworth proposes an expedition to investigate the anomaly after beating his rival, Wernstrom, in a game of Deathball. When Bender explores the anomaly, his touch causes it to emit a wave that sends him. Farnsworth and Wernstrom discover that living beings can pass through the anomaly. The two plan to another expedition but are rejected in favor of a military assault on the anomaly led by Zapp Brannigan. Meanwhile, both Fry and Bender begin to feel lonely, Fry sneaks aboard Zapps ships lint cabinet just before the ship takes off so that he can find solace on the other side of the anomaly. Bender attempts suicide, only to be approached by the League of Robots, Bender becomes a very prestigious member due to his perceived hatred of humans, although Calculon suspects that Bender is deceiving them. Fry enters the anomaly while Kif is killed during Zapps unsuccessful plan of attack, on the other side of the anomaly, Fry comes across a colossal, one-eyed, tentacled creature, which begins forcing its appendages through the anomaly. The tentacles attack everyone in the universe, and nothing can stop them since they are made of electro-matter that can only be harmed by other electro-matter, Fry returns to Earth with a tentacle attached to the back of his neck and tells everyone to love the tentacle. The tentacles attach themselves to everyone, causing their victims to fall in love with it. With the monsters influence spreading, Fry becomes the pope of a new religion established to worship the tentacles, Bender, meanwhile, believes that the League of Robots should uphold a strict anti-humans policy. However, when he assists his friends in eluding the tentacles by hiding them in another members leg, when Calculon calls his bluff about hating humans, he challenges Calculon to a duel in which he cheats. This confrontation results in the loss of Calculons arm and severe damage to much of the city, Calculon is outraged by Benders behavior and resigns from the League, making Bender its new leader. Leela ends up the last person in the universe unattached to a tentacle, after Zapp and she examines a tentacle fragment and discovers that they are actually reproductive organs, revealing this to everyone at a universal religious gathering. The creature, named Yivo, admits that mating with everyone in the universe was its original intention, as a sign of good faith, Yivo resurrects Kif, who is displeased to learn that Zapp lured the lonely Amy into sleeping with him. Yivo asks to begin the relationship anew and removes its tentacles from everyone, Yivo takes everyone in the universe out on a date at the same time, which goes extremely well
7.
History of the Theory of Numbers
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History of the Theory of Numbers is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, the central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned, this was apparently going to be the topic of a fourth volume that was never written. Volume 1 - Divisibility and Primality -486 pages Volume 2 - Diophantine Analysis -803 pages Volume 3 - Quadratic and Higher Forms -313 pages Carmichael, recent Publications, Reviews, History of the Theory of Numbers. Volume I, Divisibility and Primality, The American Mathematical Monthly,26, 396–403, doi,10. 2307/2971917, ISSN 0002-9890 Carmichael, Robert D. New York, Dover Publications, ISBN 978-0-486-44232-7, MR0245499, Zbl 1214.11001 Dickson, Leonard Eugene, History of the theory of numbers. Vol. II, Diophantine analysis, New York, Dover Publications, ISBN 978-0-486-44233-4, MR0245500, Zbl 1214.11002 Dickson, Leonard Eugene, History of the theory of numbers
8.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
9.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0