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Alphonse De Polignac
Alphonse de Polignac (1826–1863) was a French mathematician. In 1849, the year he was admitted to Polytechnique, he made what's known as Polignac's conjecture: From p. 400: ''"1er ''Théorème.'' Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières … "'' (1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways … ) :For every positive integer ''k'', there are infinitely many prime gaps of size 2''k''. The case ''k'' = 1 is the twin prime conjecture. He also conjectured Romanov's theorem. His father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown (1830). See also *de Polignac's formula *Polignac family The House of Polignac is the name of an ancient and powerful French noble family that took its name from the '' château de Polignac'', of which they had been ''sieurs'' since Caro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polignac's Conjecture
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive even number ''n'', there are infinitely many prime gaps of size ''n''. In other words: There are infinitely many cases of two consecutive prime numbers with difference ''n''. Although the conjecture has not yet been proven or disproven for any given value of ''n'', in 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size ''n'' for some value of ''n'' < 70,000,000. Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600. As of April 14, 2014, one year after Zhang's announcement, according to the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''''n'' every prime can be written as :p_ = 2 + \sum_^n g_i. Simple observations The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime Conjecture
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Romanov's Theorem
In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base , the set of numbers that are the sum of a prime and a positive integer power of has a positive lower asymptotic density. Statement Romanov initially stated that he had proven the statements "In jedem Intervall (0, x) liegen mehr als ax Zahlen, welche als Summe von einer Primzahl und einer k-ten Potenz einer ganzen Zahl darstellbar sind, wo a eine gewisse positive, nur von k abhängige Konstante bedeutet" and "In jedem Intervall (0, x) liegen mehr als bx Zahlen, weiche als Summe von einer Primzahl und einer Potenz von a darstellbar sind. Hier ist a eine gegebene ganze Zahl und b eine positive Konstante, welche nur von a abhängt". These statements translate to "In every interval (0,x) there are more than \alpha x numbers which can be represented as the sum of a prime number and a -th power of an integer, where \al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jules De Polignac
Jules Auguste Armand Marie de Polignac, Count of Polignac (; 14 May 178030 March 1847), then Prince of Polignac, and briefly 3rd Duke of Polignac in 1847, was a French statesman and ultra-royalist politician after the Revolution. He served as prime minister under Charles X, just before the July Revolution in 1830 that overthrew the senior line of the House of Bourbon. It is admitted he is the one responsible for the colonisation of Algeria by France as he led the July 1830 expedition to conquer Algeria. Early life Born in Versailles, Jules was the younger son of Jules, 1st Duke of Polignac, and Gabrielle de Polastron, a confidante and favourite of Queen Marie-Antoinette. Due to his mother's privileged position, the young Jules was raised in the environment of the court of Versailles, where his family occupied a luxurious suite of thirteen rooms. His sister, Aglaé, was married to the duc de Guîche at a young age, helping to cement the Polignac family's position as one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Polignac's Formula
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime ''p'' that divides the factorial ''n''!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac. Statement For any prime number ''p'' and any positive integer ''n'', let \nu_p(n) be the exponent of the largest power of ''p'' that divides ''n'' (that is, the ''p''-adic valuation of ''n''). Then :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor, where \lfloor x \rfloor is the floor function. While the sum on the right side is an infinite sum, for any particular values of ''n'' and ''p'' it has only finitely many nonzero terms: for every ''i'' large enough that p^i > n, one has \textstyle \left\lfloor \frac \right\rfloor = 0. This reduces the infinite sum above to :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor \, , where L = \lfloor \log_ n \rfloor. Example For ''n'' = 6, one has 6! = 720 = 2^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polignac Family
The House of Polignac is the name of an ancient and powerful French noble family that took its name from the '' château de Polignac'', of which they had been ''sieurs'' since Carolingian times. Agnatically, ruling family of Monaco represents the collateral branch of the House of Polignac. History In 1385, the male line became extinct, but the heiress married Guillaume, sire de Chalancon, who assumed the name and the coat of arms of Polignac family. Jules de Polignac (1746–1817) became the first Duke of Polignac in 1780. Notable family members * Melchior de Polignac (1661–1742), French diplomat, Roman Catholic cardinal and neo-Latin poet * Jules de Polignac (1746–1817), became the first Duke of Polignac * Gabrielle de Polastron, duchesse de Polignac (1749–1793), wife of the first Duke of Polignac * Jules, prince de Polignac (1780–1847), promulgator of the July Ordinances * Alphonse de Polignac (1826–1863), French mathematician and number theorist * Camille Arm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1826 Births
Eighteen or 18 may refer to: * 18 (number), the natural number following 17 and preceding 19 * one of the years 18 BC, AD 18, 1918, 2018 Film, television and entertainment * ''18'' (film), a 1993 Taiwanese experimental film based on the short story ''God's Dice'' * ''Eighteen'' (film), a 2005 Canadian dramatic feature film * 18 (British Board of Film Classification), a film rating in the United Kingdom, also used in Ireland by the Irish Film Classification Office * 18 (''Dragon Ball''), a character in the ''Dragon Ball'' franchise * "Eighteen", a 2006 episode of the animated television series ''12 oz. Mouse'' Music Albums * ''18'' (Moby album), 2002 * ''18'' (Nana Kitade album), 2005 * '' 18...'', 2009 debut album by G.E.M. Songs * "18" (5 Seconds of Summer song), from their 2014 eponymous debut album * "18" (One Direction song), from their 2014 studio album ''Four'' * "18", by Anarbor from their 2013 studio album '' Burnout'' * "I'm Eighteen", by Alice Cooper common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1863 Deaths
Events January–March * January 1 – Abraham Lincoln signs the Emancipation Proclamation during the third year of the American Civil War, making the abolition of slavery in the Confederate states an official war goal. It proclaims the freedom of 3.1 million of the nation's four million slaves and immediately frees 50,000 of them, with the rest freed as Union armies advance. * January 2 – Lucius Tar Painting Master Company (''Teerfarbenfabrik Meirter Lucius''), predecessor of Hoechst, as a worldwide chemical manufacturing brand, founded in a suburb of Frankfurt am Main, Germany. * January 4 – The New Apostolic Church, a Christian and chiliastic church, is established in Hamburg, Germany. * January 7 – In the Swiss canton of Ticino, the village of Bedretto is partly destroyed and 29 killed, by an avalanche. * January 8 ** The Yorkshire County Cricket Club is founded at the Adelphi Hotel, in Sheffield, England. ** American Civil War &ndash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
