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[ "In order for to be prime , it is necessary that should be prime .", "In order for 2p â ' 1 to be prime , it is necessary that p itself be prime ." ]
[ "This means there are 47 perfect numbers known , the largest being 243,112,609 ( 243,112,609 1 ) with 25,956,377 digits .", "The largest of these is 243,112,608 Ã -- ( 243,112,609 â ' 1 ) with 25,956,377 digits ." ]
[ "Its divisors ( other than the number itself : 6 ) are 1 , 2 , and 3 and 1 + 2 + 3 equals 6 .", "For example , the digital root of 8128 = 1 , since 8 + 1 + 2 + 8 = 19 , 1 + 9 = 10 , and 1 + 0 = 1 . The reason that this digital root does n't work with the perfect number 6 is because it only works with perfect numb...
[ "Therefore , every Mersenne prime will yield a distinct even perfect number -- there is a concrete one-to-one association between even perfect numbers and Mersenne primes .", "Thus , there is a one-to-one relationship between even perfect numbers and Mersenne primes ; each Mersenne prime generates one even perfec...
[ "Even perfect numbers have a very precise form ; odd perfect numbers are rare , if indeed they do exist .", "All even perfect numbers have a very precise form ; odd perfect numbers either do not exist or are rare ." ]
[ "Also , any even perfect number except the first one is the sum of the first 2 ( n 1 ) \\/ 2 odd cubes :", "Furthermore , any even perfect number except the first one is the ( ( 2p +1 ) \\/ 3 ) th centered nonagonal number as well as the sum of the first 2 ( pâ ' 1 ) \\/ 2 odd cubes :" ]
[ "Till September 2009 , only 47 Mersenne primes are known .", "As of 2010 , 47 Mersenne primes and therefore 47 even perfect numbers are known ." ]
[ "It is not known whether there are any odd perfect numbers .", "It is unknown whether there are any odd perfect numbers ." ]
[ "Prime numbers of the form 2n 1 are known as Mersenne primes , after the seventeenth-century monk Marin Mersenne , who studied number theory and perfect numbers .", "Prime numbers of the form 2p â ' 1 are known as Mersenne primes , after the seventeenth-century monk Marin Mersenne , who studied number theory and ...
[ "Six is the first perfect number . Its divisors ( other than the number itself : 6 ) are 1 , 2 , and 3 and 1 + 2 + 3 equals 6 .", "The first perfect number is 6 , because 1 , 2 , and 3 are its proper positive divisors , and 1 + 2 + 3 = 6 ." ]
[ "Other perfect numbers include 28 , 496 and 8128 .", "This is followed by the perfect numbers 496 and 8128 ." ]
[ "Since any even perfect number has the form 2n 1 ( 2n 1 ) , it is a triangular number , and , like all triangular numbers , it is the sum of all natural numbers up to a certain point ; in this case : 2n 1 .", "Like all triangular numbers , it is the sum of all natural numbers up to a certain point ; in this case ...
[ "The electrons interact with the atoms that make up the specimen , producing signals that contain information about both the shape and other details - for instance the atomic structure , or the conductivity .", "The electrons interact with the atoms that make up the sample producing signals that contain informati...
[ "The scanning electron microscope ( SEM ) is a type of electron microscope that makes an image by moving a narrow electron beam across the surface of a specimen .", "A scanning electron microscope ( SEM ) is a type of electron microscope that images a sample by scanning it with a high-energy beam of electrons in ...
[ "Fritz Herkenrath ( born 9 September , 1928 ) is a former German football player .", "Friedrich `` Fritz '' Herkenrath ( born September 9 , 1928 in Cologne ) is a former football goalkeeper for Germany at the 1958 FIFA World Cup ." ]
[ "Saint-Pierremont , Aisne is a commune .", "Saint-Pierremont is a commune in the Aisne department in Picardy in northern France ." ]
[ "Kosuke Fukudome ( born April 26 , 1977 ) is a Japanese baseball player . He currently plays as an outfielder for the Chicago Cubs of Major League Baseball ( MLB ) .", "Kosuke Fukudome ( pronounced KOH-skay foo-koo-DOUGH-may ) is a Japanese outfielder for the Chicago Cubs ." ]
[ "Fukudome formerly played for the Chunichi Dragons of the Central League from 1998 to 2007 .", "The Chunichi Dragons drafted Fukudome in 1998 in the first round as a shortstop ." ]
[ "Natural gas is a mixture .", "Natural gas is commercially extracted from oil fields and natural gas fields . Gas extracted from oil wells is called casinghead gas or associated gas ." ]
[ "Issues that it involves itself in are judicial nominations , church\\/state separation , civil rights , voting rights for Washington , DC in the U.S. Congress and equal rights for lesbian , gay , bisexual , and transgendered ( LGBT ) people and promotion of civic participation .", "PFAW has been active in recent...
[ "PFAW says about its ' purpose on its ' website People For the American Way is dedicated to making the promise of America real for every American : Equality .", "People For the American Way is dedicated to making the promise of America real for every American : Equality ." ]
[ "JosÃ de Acosta ( September\\/October 1539 -- February 15 1600 ) , was a Spanish 16th-century Jesuit missionary and naturalist in Latin America .", "JosÃ de Acosta ( September\\/October 1539 â `` February 15 , 1600 ) was a Spanish 16th-century Jesuit missionary and naturalist in Latin America ." ]
[ "Malad City is a city of Idaho in the United States .", "Malad City ( also commonly known as Malad ) is the only city in and the county seat of Oneida County , Idaho , United States ." ]
[ "1504 was a year of the Gregorian calendar , that started on a Friday .", "Year 1504 ( MDIV ) was a leap year starting on Monday ( link will display the full calendar ) of the Julian calendar ." ]
[ "Bobby Convey ( born 27 May 1983 ) is an American soccer player . He plays for San Jose Earthquakes .", "Robert Francis `` Bobby '' Convey ( born May 27 , 1983 in Philadelphia , Pennsylvania ) is an American soccer player who plays for San Jose Earthquakes in Major League Soccer ." ]
[ "Deauville is a commune . It is found in the region Basse-Normandie in the Calvados department in the northwest of France .", "Deauville is a commune in the Calvados department in the Basse-Normandie region in northwestern France ." ]
[ "Koblenz is the base of the Campus Koblenz of the University of Koblenz-Landau and the seat of the Mayen-Koblenz municipality .", "The campus Koblenz of University of Koblenz and Landau is located in the city ." ]
[ "Vernon is a commune in the ArdÃ che dÃ partement in southern France .", "Vernon is a commune in the ArdÃ che department in southern France ." ]
[ "Rhadamistus ( other names Ghadam or Radamisto ) was an Iberian prince who reigned in Armenia from 51 to 53 and 54 to 55 CE .", "Rhadamistus ( also known as Ghadam or Radamisto ) was an Iberian prince who reigned in Armenia from 51 to 53 and 54 to 55 CE ." ]
[ "Divaldo Pereira Franco or simply , Divaldo Franco is a Brazilian medium , born in Feira de Santana , Bahia , on May 5th , 1927 .", "Divaldo Pereira Franco ( born May 5 , 1927 , Feira de Santana , Brazil ) is a prominent Spiritist speaker and medium ." ]
[ "Samuel Huntington ( July 16 , 1731 - January 5 , 1796 ) was an American jurist , statesman , and revolutionary leader from Connecticut .", "Samuel Huntington ( - January 5 , 1796 ) was a jurist , statesman , and Patriot in the American Revolution from Connecticut ." ]
[ "He served as a delegate to the Continental Congress where he signed the Declaration of Independence , as Governor of Connecticut , and later as the first President of the United States in Congress Assembled , that is , the presiding officer of the Congress of the Confederation , the sole governing body of the firs...
[ "During an Avatar : The Last Airbender episode , titled `` The Tales of Ba Sing Se '' ( episode 15 of the second season ) , which comprises several small stories about the main characters , there is a segment titled , `` The Tale of Iroh . ''", "During an Avatar : The Last Airbender episode , titled `` The Tales ...
[ "Mako Iwamatsu , born Makoto Iwamatsu ( 10 December 1933 â `` 21 July 2006 ) , was a Japanese-American actor .", "Mako Iwamatsu , born Makoto Iwamatsu ( December 10 , 1933 â `` July 21 , 2006 ) , was a Japanese-American actor ." ]
[ "Mako died in Somis , California on 21 July 2006 , aged 72 , from esophageal cancer .", "Mako died in Somis , California on July 21 , 2006 , aged 72 , from esophageal cancer ." ]
[ "Brig is a part of the canton of Valais , Switzerland .", "The district of Brig is a district in the canton of Valais , Switzerland ." ]
[ "There are no thru roads to Juneau .", "The only airport in Juneau is Juneau International Airport ." ]
[ "Goms is a district of the canton of Valais in Switzerland .", "The district of Goms is a district of the canton of Valais in Switzerland ." ]
[ "In the Lord of the Rings , however , elves were described as human sized ( and if anything taller ) , hence changing the description of elves .", "This prompts Grimm to assume a tripartite division of light elves , dark elves and black elves , of which only the latter are identical with dwarves , while the dark ...
[ "Moonspell 's first EP was Under the Moonspell , which was released in 1994 .", "Moonspell is a Portuguese heavy metal band from Brandoa , Lisbon . Formed in 1992 , the group released their first EP Under the Moonspell in 1994 , a year before the release of their first album Wolfheart ." ]
[ "He spent most of his childhood in Trieste .", "They were often touring during his childhood , and he was brought up in Trieste for the most part ." ]
[ "During World War I , Busoni lived first in Bologna , where he was director of the conservatory , and later in ZÃ 1\\/4 rich .", "During World War I , Busoni lived first in Bologna , where he directed the conservatory , and later in ZÃ 1\\/4 rich ." ]
[ "Ferruccio Busoni was born in Empoli in Tuscany in Italy . His parents were both musicians .", "Ferruccio Busoni was born in Empoli in Tuscany in Italy , the only child of two professional musicians ." ]
[ "Ferruccio Busoni ( born 1 April ; died 27 July 1924 ) was an Italian composer , pianist , teacher and conductor .", "Ferruccio ( Dante Michelangelo Benvenuto ) Busoni ( April 1 , 1866 -- July 27 , 1924 ) was an Italian composer , pianist , editor , writer , piano and composition teacher , and conductor ." ]
[ "Robert Johnson , born Robert Leroy Johnson ( May 8 , 1911 â `` August 16 , 1938 ) is one of the most famous Delta blues musicians .", "Robert Leroy Johnson ( May 8 , 1911 â `` August 16 , 1938 ) was an American blues singer and musician ." ]
[ "Eleven grammophone records were made by Robert Johnson with Vocalion during his lifetime , and a twelfth record was made once he had died .", "Eleven Johnson 78s were released on the Vocalion label during his lifetime , with a twelfth issued posthumously . All songs are copyrighted to Robert Johnson , and his es...
[ "He was born in Hazlehurst , Mississippi . He was the 11th child of his mother Julia Major Dodds , and 10th child of his father Charles Dodds .", "Julia herself left Hazlehurst with baby Robert , but after some two years , sent him to live in Memphis with Dodds , who had changed his name to Charles Spencer ." ]
[ "Backlash ( 2004 ) was a professional wrestling pay-per-view event produced by World Wrestling Entertainment ( WWE ) , which took place on April 18 , 2004 at the Rexall Place in Edmonton , Alberta , Canada .", "Backlash ( 2004 ) was the sixth annual Backlash professional wrestling pay-per-view ( PPV ) event produ...
[ "Donegal is a town in County Donegal , Republic of Ireland .", "Donegal , usually referred to as Donegal Town is a town in County Donegal , Ireland ." ]
[ "Such a family has not been commonly recognized by taxonomists are the plants involved were usually included in the family Xanthorrhoeaceae .", "Such a family has not been commonly recognized by taxonomists : the plants involved were usually included in the family Xanthorrhoeaceae ." ]
[ "The family is native to Australia , and contains 16 species in four genera .", "The family is endemic to Australia , and comprises 16 species in four genera ." ]
[ "FregiÃ court is a municipality of the district of Porrentruy in the canton of Jura in Switzerland .", "FregiÃ court is a village and a former municipality in the district of Porrentruy in the canton of Jura in Switzerland ." ]
[ "Couvrelles is a commune . It is found in the region Picardie in the Aisne department in the north of France .", "Couvrelles is a commune in the Aisne department in Picardy in northern France ." ]
[ "It belongs to the Carme group , made up of non-spherical retrograde moons orbiting Jupiter at a distance ranging between 23,000,000 and 24,000,000 km and at an inclination of about 165 Â .", "It belongs to the Carme group , made up of irregular retrograde moons orbiting Jupiter at a distance ranging between 23 a...
[ "is about 1 kilometre in diameter , and orbits Jupiter at an average distance of 23,858,000 km in 752.839 days , at an inclination of 165 Â to the ecliptic ( 165 Â to Jupiter 's equator ) , with an eccentricity of 0.276 .", "is about 1 kilometre in diameter , and orbits Jupiter at an average distance of 23,858 Mm...
[ "It was found by a team of astronomers from the University of Hawaii led by Scott S. Sheppard in 2003 .", "It was discovered by a team of astronomers from the University of Hawaii led by Scott S. Sheppard in 2003 ." ]
[ "The feathers of the flightless Velociraptor may have been used for display , for covering their nests while brooding .", "The feathers of the flightless Velociraptor may have been used for display , for covering their nests while brooding , or for added speed and thrust when running up inclined slopes ." ]
[ "Therefore , the feathers in the ancestors of Velociraptor had another function .", "The fact that the ancestors of Velociraptor were feathered and possibly capable of flight long suggested to paleontologists that Velociraptor bore feathers as well , since even flightless birds today retain most of their feathers...
[ "Turner and colleagues interpreted the presence of feathers on Velociraptor as evidence that the ancestors of dromaeosaurids could fly , making Velociraptor and other large members of this family secondarily flightless .", "Turner and colleagues interpreted the presence of feathers on Velociraptor as evidence aga...
[ "Most of the continental crust is much older .", "In contrast , the bulk of the continental crust is much older ." ]
[ "Such old continental crust and the mantle below it are less dense than other places in the earth .", "Such old continental crust and the underlying mantle asthenosphere are less dense than elsewhere in the earth and so are not readily destroyed by subduction ." ]
[ "The Earth 's mantle and crust formed about 100 million years after the formation of the planet , about 4.6 billion years ago .", "Partly by analogy to what is known about our Moon , Earth is considered to have differentiated from an aggregate of planetesimals into its core , mantle and crust within about 100 mil...
[ "The upper part of the mantle is made up of peridotite , a rock denser than rocks common in the crust .", "The upper part of the mantle is composed mostly of peridotite , a rock denser than rocks common in the overlying crust ." ]
[ "The temperature of the crust increases with depth . Where the crust meets the mantle the temperatures can be between 200 Â C ( 392 Â F ) to 400 Â C ( 752 Â F ) .", "The temperature of the crust increases with depth , reaching values typically in the range from about 200 Â C ( 392 Â F ) to 400 Â C ( 752 Â F ) at ...
[ "A few zircons at least 4.3 billion years old have been found in the Narryer Gneiss Terrane .", "A few zircons with ages as great as 4.3 billion years have been found in the Narryer Gneiss Terrane ." ]
[ "The lithosphere is broken up into tectonic plates that can move .", "Because of convection in the underlying plastic ( although non-molten ) upper mantle and asthenosphere , the lithosphere is broken into tectonic plates that move ." ]
[ "The making of new continental crust is linked to times of major orogeny or mountain building . This happens at the same time as the formation of the supercontinents such as Rodinia , Pangaea and Gondwana .", "Formation of new continental crust is linked to periods of intense orogeny or mountain building ; these ...
[ "The crust forms in part by the coming together of island arcs including granite and metamorphic fold belts . They are kept together partly by the using up of the mantle below the crust , which makes a mantle on which the crust can float .", "The crust forms in part by aggregation of island arcs including granite...
[ "The continental crust is thicker , and the oceanic crust is thinner .", "Because the top of the continental crust is above that of the oceanic , water runs off the continents and collects above the oceanic crust . The continental crust and the oceanic crust are sometimes called sial and sima respectively ." ]
[ "They are kept together partly by the using up of the mantle below the crust , which makes a mantle on which the crust can float .", "Both the continental and oceanic crust `` float '' on the mantle ( see isostasy ) ." ]
[ "Most crustal rocks formed before 2.5 billion years ago are in cratons .", "Most crustal rocks formed before 2.5 billion years ago are located in cratons ." ]
[ "The Earth 's crust is less than 1 % of Earth 's volume .", "Earth 's crust occupies less than 1 % of Earth 's volume ." ]
[ "The crust has two different parts . One is the continental crust ( under the land ) and the other is oceanic crust ( under the ocean ) .", "The oceanic crust of the sheet is different from its continental crust ." ]
[ "The oldest oceanic basalt crust today is only about 200 million years .", "The Earth has probably always had some form of basaltic crust , but the age of the oldest oceanic crust today is only about 200 million years ." ]
[ "Below the crust is the mantle .", "The crust is underlain by the mantle ." ]
[ "The oldest continental crustal rocks on Earth are between 3.7 to 4.28 billion years old . These have been found in the Narryer Gneiss Terrane in Western Australia , in the Acasta Gneiss in the Northwest Territories on the Canadian Shield , and on the Fennoscandian Shield .", "The oldest continental crustal rocks...
[ "The crust and the upper mantle make up the lithosphere .", "The crust and underlying relatively rigid uppermost mantle make up the lithosphere ." ]
[ "Hexham Abbey is an Anglican church in the town of Hexham , Northumberland , dedicated to St Andrew .", "Hexham Abbey is a place of Christian worship dedicated to St Andrew and located in the town of Hexham , Northumberland , in northeast England ." ]
[ "There has been a church on this site for over 1300 years since Etheldreda , Queen of Northumbria , made a grant of lands c. 674 .", "There has been a church on the site for over 1300 years since Etheldreda , Queen of Northumbria made a grant of lands to Wilfrid , Bishop of York c. 674 ." ]
[ "The Benedictine abbey was built almost entirely of stones from nearby Roman ruins . The Saxon crypt still remains ; as does a frith stool , a 7th\\/8th century ` cathedra ' or throne .", "Of Wilfrid 's Benedictine abbey , which was constructed almost entirely of material salvaged from nearby Roman ruins , the Sa...
[ "The east end was rebuilt in 1860 , and the rest of the Abbey was largely rebuilt around 1900 .", "The east end was rebuilt in 1860 ." ]
[ "The choir , north and south transepts and the cloisters date from this period .", "The choir , north and south transepts and the cloisters , where canons studied and meditated , date from this period ." ]
[ "The present church is from that period ( c. 1170 â `` 1250 ) , in the Early English ( Gothic ) style of architecture .", "The current church largely dates from that period ( c. 1170 â `` 1250 ) , in the Early English style of architecture ." ]
[ "The nave was re-consecrated in 1908 .", "The nave was re-consecrated on August 8 , 1908 ." ]
[ "His son Eilaf II completed the work probably building in the Norman style .", "His son Eilaf II completed the work , probably building in the Norman style ." ]
[ "In Norman times , the old abbey was replaced by an Augustinian priory .", "In Norman times , Wilfrid 's abbey was replaced by an Augustinian priory ." ]
[ "This mammoth project involved re-building the nave , whose walls include some of the earlier church .", "This mammoth project involved re-building the nave , whose walls incorporate some of the earlier church and the restoration of the choir ." ]
[ "F.C. Tokyo is a football club which plays in Japan .", "F.C. Tokyo is a Japanese association football club playing in J. League Division 2 . Its hometown is Tokyo prefecture ." ]
[ "The World Conservation Union ( IUCN ) has worked out that endangered species are 40 % of all organisms .", "The International Union for Conservation of Nature ( IUCN ) has calculated the percentage of endangered species as 40 percent of all organisms based on the sample of species that have been evaluated throug...
[ "He also helped start the cognitive revolution in psychology through his review of B. F. Skinner 's Verbal Behavior .", "In 1959 , Chomsky published a widely influential review of B. F. Skinner 's theoretical book Verbal Behavior . In this review and other writings , Chomsky broadly and aggressively challenged th...
[ "According to the Arts and Humanities Citation Index in 1992 , Chomsky was cited as a source more often than any other living scholar during the 1980 â `` 1992 time period .", "According to the Arts and Humanities Citation Index in 1992 , Chomsky was cited as a source more often than any other living scholar from...
[ "His natural approach to the study of language also changed the philosophy of language and mind .", "His naturalistic approach to the study of language has influenced the philosophy of language and mind ." ]
[ "Because of that , Chomsky became more widely known for his media criticism and politics . He is a key intellectual figure within the left wing of United States politics .", "In the political arena , Chomsky 's status as a key intellectual figure within the American left wing has also resulted in criticism and le...
[ "Chomsky created the theory of generative grammar .", "In the 1950s , Chomsky began developing his theory of generative grammar , which has undergone numerous revisions and has had a profound influence on linguistics ." ]
[ "Avram Noam Chomsky ( born December 7 , 1928 ) is an American linguist , philosopher , political activist , author , and lecturer .", "Avram Noam Chomsky ( ; born December 7 , 1928 ) , known as Noam Chomsky , is an American linguist , philosopher , cognitive scientist , and political activist ." ]
[ "Katsuyuki Kawachi ( born 27 April 1955 ) is a former Japanese football player .", "Katsuyuki Kawachi ( born April 27 , 1955 ) is a former Japanese football player ." ]
[ "Cellular division occurs along a single axis in these bacteria , and thus they grow in chains or pairs , hence the name from Greek Ï Ï Ï Î Ï Ï Î Ï streptos , meaning easily bent or twisted , like a chain ( twisted chain ) .", "Cellular division occurs along a single axis in these bacteria , and thus they grow in...
[ "Streptococci are oxidase and catalase-negative , and many are facultative anaerobes .", "Streptococci are oxidase - and catalase-negative , and many are facultative anaerobes ." ]
[ "Ryuta Hara ( born 19 April 1981 ) is a former Japanese football player .", "Ryuta Hara is a retired Japanese footballer ." ]
[ "Ann Coulter ( born December 8 , 1961 ) is from the United States .", "Ann Hart Coulter ( born December 8 , 1961 ) is an American lawyer , conservative social and political commentator , author , and syndicated columnist ." ]

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