81. The Prime Glossary: Catalan's Problem Portrait of Eugène catalan Eugène catalan. JOC/EFR August 2001 http://primes.utm.edu/glossary/page.php/CatalansProblem.html

Catalan's problem (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index In 1844, the Belgian mathematician Eugène Charles Catalan conjectured that 8 and 9 are the only pair of consecutive powers. That is, the Catalan equation for primes p and q and positive integers x and y x p y q has only the one solution: In 1976 R. Tijdeman took the first major step toward showing this by proving that for any solution y q is less than e^e^e^e^730 (a huge number!) [Guy94]. Since then this bound has been reduced many times, and we now know that the larger of p and q is at most 7.78 and the smaller is at least 10 One of the most interesting results about solutions to this problem is that any solutions other than the pair ( p q ) = (2,3) must satisfy both of: p q mod q q p 1 (mod p That is p is a Wieferich prime base q , and q is a Wieferich prime base p On April 18, 2002, Preda Mihailescu sent a manuscript to several mathematicians with a proof of the entire conjecture accompanied by an expository analysis by Yuri Bilu. It is expected that as soon as this work is completely reviewed by other mathematicians that Catalan's conjcture has been proved! Fermat -Catalan equation Solutions to Catalan's conjecture and Fermat's Last Theorem are special cases of the Fermat-Catalan equation x p y q z r Where x y z are positive, coprime integers and the exponents are all primes with

82. Catalan, Eugène Charles (1814-1894) -- From Eric Weisstein's World Of Scientifi Nationality , Belgian v. catalan, Eugène Charles (18141894), Belgianmathematician who founded Nouvelle correspondence mathématique in 1874. http://scienceworld.wolfram.com/biography/Catalan.html

Branch of Science Mathematicians Nationality Belgian Belgian mathematician who founded in 1874. His name is also associated with the Catalan numbers which are fundamental objects in combinatorial enumeration, and Catalan's conjecture a difficult problem in Diophantine equation theory which states that 8 and 9 are the only nontrivial consecutive powers. References Altshiller-Court, N. Modern Pure Solid Geometry. New York: Chelsea, p. 297, 1979. Author: Eric W. Weisstein

83. 80mundos Translate this page en Todas las Categorías. http://www.80mundos.com/busqueda.asp?OPCION=2&TEXTO=IONESCO.-EUGENE

84. VirtuallyThere Weather Eugene, Oregon English Conditions and Forecast for eugene, Oregon. http://www.wunderground.com/auto/virtuallythere/OR/Eugene.html

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85. IAS Supporters Paula Emery Michael Glavin Elizabeth Elson Paul Glavin Greg Hall Wendy Hyman BrookeLehman Cindy Milstein Chuck Morse John Petrovato eugene Rodgriguez Peter http://flag.blackened.net/ias/supporters.htm

Introduction: English Español Português ... How To Help IAS Supporters Great Books Apply for a Grant Who is the IAS? Contact the IAS ... Site Index IAS Supporters The following groups and individuals have generously donated or pledged a donation to the IAS's 2002 fundraising campaign. Ashanti Alston David Berberick Dan Chodorkoff Miranda Edison David Eisen Paula Emery Paul Glavin Greg Hall Wendy Hyman Brooke Lehman Cindy Milstein Chuck Morse John Petrovato Eugene Rodgriguez Peter Stone Patti Stanko Michael Winslow Go to How to Help the IAS in 2000 See listings of IAS supporters. The IAS's Supporters The following groups and individuals donated generously to the IAS's 2000 fundraising campaign. Individuals: Anonymous I and II Randall Amster Espíritu Beothuk Phil Billingsley Jon Bekken David Berberick Craig Bolton Manuel Callahan Eric Chester Salvator Engle-Dimauro Deborah Engle-Dimauro Chris Dodge Miranda Edison Howard Ehrlich David Eisen Paula Emery Tony Epiceno

86. Palace Of Catalan Music (Palau De La Musica Catalana) - Reviews And Information Palace of catalan Music (Palau de la Musica catalana), http://smarterliving.tripadvisor.com/Attraction_Review-g187497-d190824-Reviews-P

writeInputField("Enter e-mail address"); Find (e.g., Boston hotels, Las Vegas, Paris art museum) in Barcelona in Spain in Europe anywhere Powered by Palace of Catalan Music (Palau de la Musica Catalana) You are here: Home Europe Spain Catalonia ... Barcelona Attractions Palace of Catalan Music (Palau de la Musica Catalana) Barcelona: Tourism Hotels Attractions Special deals Maps, etc. Palace of Catalan Music (Palau de la Musica Catalana) View all Barcelona attractions C/Sant Francesc de Paula 2 Eixample Barcelona Spain Tel: 93 295 72 00 URL: http://www.palaumusica.org This music hall features classical guitar, orchestra and occasional flamenco performances. Special offers on Barcelona hotels and vacations (1-4 of 509) Sponsored links Low prices on Barcelona hotels - Click Here - Orbitz Low prices on Barcelona hotels - Click Here - Expedia.com Low prices on Barcelona hotels - Click Here - Travelocity Special Vacation Packages to Barcelona - Expedia.com - Click Here ... View all 509 Barcelona vacation packages Featured airfares to Barcelona (arriving in Barcelona, Spain) Departing from: Chicago, IL/O'Hare (ORD)

87. Catalan¡¦s ²q·Q The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set. http://sylvester.math.nthu.edu.tw/d2/appreciation/Catalan¡¦s ²q·Q.htm

Catalan¡¦s ²q·Q ²q·Q¡G¡u¦b©Ò¦³¾ã¼Æªº©Ò¦³¦¸¤è©Ò²Õ¦¨ªº¼Æ¦C¤¤¡A°ß¤@ªº³sÄò¾ã¼Æ¥u¦³ ©M ¡v ¦p¤µ¡A¼w°ê Paderborn ¤j¾Çªº Perda Mihailescu ®õ¹¥§¨È²q·Q ¡K¡C¦b³o¼Æ¦C·í¤¤¡A ©M ¤£¦ý¬O¾ã¼Æªº¦¸¤è¼Æ¡A¤]¬O­Ó³sÄò¾ã¼Æ¡C ¦~¡A¤ñ§Q®É¼Æ¾Ç®a Eugene Charles Catalan (1814~1894) ´£¥X²q·Q¡G¡u¦b©Ò¦³¾ã¼Æªº©Ò¦³¦¸¤è©Ò²Õ¦¨ªº¼Æ¦C¤¤¡A°ß¤@ªº³sÄò¾ã¼Æ¥u¦³ ©M ¡v ¦pªG­n¸Ñ¨M Catalan ³£¤j©ó ¦³½ìªº¬O¡A¦b Catalan ²q·Q´£¥X«e ¦~¡A Levi ben Gerson (1288~1344) ¤w¸gÒ©ú¥X¦b©Ò¦³¾ã¼Æªº¥­¤è©M¥ß¤è¼Æ¤¤¡A¬Û®t¬° ªº¾ã¼Æ¥u¦³ ©M ¡C ¦~¡A¯¦èÄõ Leiden ¤j¾Çªº Robert Tijdeman p q ¦~¡A Mihailescu Double Wieferich Primes ¡C³o¹ï½è¼Æ¿í¦u¤U¦C³W«h¡G °£¥H ¾l ¡A °£¥H¤]¾l ¡CÂù Wieferich ½è¼Æ«D±`¤§¨u¨£¡A¥Ø«e¥u§ä¨ì¤»­Ó¡G ¡B ¡B ¡B ¡B ¡B ¡A¦Ó¨ä¤¤¨S¦³¤@­Óº¡¨¬ Cantalan ²q·Qªº¦¡¤l¡C Mihai lescu Mihailescu »¡¡A¥LªºÒ©ú§Q¥Î¨ì¥ý«eÂù Wieferich ½è¼Æªºµ²ªG¡C ¦Û¥Ø«e¬°¦¹¡A Mihailescu ªºÒ©ú¨«Dµ´¹ïªº¥i«H¡A¦ý«o¦³¤Q¤À¥i¯àªº¼x¥ü¡Aªk°ê Talence Bordeaux I ¤j¾Çªº Yuri F. Bilu ¤w¤ÀªR Mihailescu ªºÒ©ú¡A¨¬°¥¦¼g¤U¥D­n¨BÆJªº·§­n¤Î¦³½ìªºµû½×¡A¥¦»¡¡G¡u§Ú½T«H Mihailescu ªºÒ©ú¬O¹ïªº¡I¡v ¤é¡A Mihailescu ¦b»X¯S°ú¥« ¥[®³¤jªF«n³¡´ä¥« ³o»ò¬Ý¨Ó¡A Cantalan ²q·Q§Y±N¨B¤J¼Æ¾Ç¤j©w²zªº·µ°ó¤¤¡I ºK¦Û ¡G Ivars Peterson 2002 ¦^¼Æ¾ÇªY½à

88. Biography-center - Letter O Any language, www.geocities.com/moondarlin/artokeeffe.html; O'Neill, eugene odur.let http://www.biography-center.com/o.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish O 134 biographies O'Brian, Patrick www.wwnorton.com/pob/bio.htm O'Brien, Margaret www.wic.org/bio/mobrien.htm O'connor, Justice supct.law.cornell.edu/supct/justices/oconnor.bio.clr.html O'Connor, Flannery www.wwnorton.com/naal/explore/oconnor.htm O'Connor, Mary Flannery cheshire.cwrl.utexas.edu/~mmaynard/voices/oconnor.html O'Hara, John www.ncteamericancollection.org/litmap/ohara_john_pa.htm O'Hara, Catherine www.northernstars.ca/actorsmno/oharabio.html O'Keeffe, Georgia www.pbs.org/wnet/americanmasters/database/okeeffe_g.html O'Keeffe, Georgia www.geocities.com/moondarlin/artokeeffe.html O'Neill, Terri library.thinkquest.org/20117/oneill.html O'Neill, Eugene odur.let.rug.nl/~usa/LIT/oneill.htm_pa.htm O'Neill, Eugene

89. Week144 These numbers were first discovered by Euler, but they're named afterEugene catalan, who discovered their relation to binary trees. http://math.ucr.edu/home/baez/week144.html

January 21, 2000 This Week's Finds in Mathematical Physics (Week 144) John Baez Since this is the first Week of the new millennium, I'd like to start with a peek into the future. Not just the next hundred or thousand years, either - I'm sick of short-term planning. No, I'd like to talk about the next few *billion* years. As you've probably all heard, if we don't do anything about it, the Sun will turn into a red giant in about 5 billion years. If we get our act together, we should have plenty of time to deal with this problem. But when planning for the far future, it's dangerous to be too parochial! Events outside our solar system can also affect us. For example, a nearby supernova could be a real bummer. It wouldn't be the first time: it seems that about 340,000 years ago there was one only 180 lightyears away. At this distance it would have been as bright as a full moon, and its X-rays and gamma rays would have stripped off the Earth's ozone layer pretty badly for a while. A closer one could be a lot worse. How we do know about this supernova? It's an interesting story. We happen to live in a region of space called the Local Bubble, about 300 lightyears across, in which the interstellar gas is hotter and 5 to 10 times less dense than the surrounding stuff. People wondered about the origin of this bubble until they studied a pulsar called Geminga about 300 lightyears away from us. Pulsars are rapidly spinning neutron stars formed by supernovae, and by studying their spin rate and the rate their spin is slowing down, you can guess when they were formed. Geminga turns out to be about 340,000 years old. It's moving away from us at a known rate, so back then it would have been 180 lightyears away - in just about the right place for a supernova to have created a shock wave forming the Local Bubble.

90. Code-Switching Bibliography Database Title Implicit Curriculum in Japanese Language Teaching Case Studies of Two HighSchool Teachers of Japanese. City eugene. University University of Oregon. http://mmetsnt.sas.upenn.edu/scripts/codeswitch/searchentry.asp?SC=Languageres&S

91. Discos-clásica Translate this page The best of Louis Moreau Gottschalk, List, eugene, Turnabout. The Catalanpiano tradition, Albeniz-Malats-Granados-Marshall-De Larrocha, no figura. http://www.rigolettocurioso.com.ar/Discos/Clásica.htm

Datos para contactarse con nosotros Titulo Intérprete Editor A Gottscalk Festival List, Eugene Turnabout Adam Giselle Bolshoi Theater Orchestra Capitol Aida (Verdi) The Rome Simphony Orchestra. Dir: Domenico Savino Vanguard Aida (Verdi) Orquesta Filarmónica de Viena. Dir: Karajan London Arias de operas (Mozart) Steber-Walter Columbia Arias favoritas Pavarotti, Luciano London Beethoven-Brahms-Mendelssohn-Schubert Trio Istomin-Stern-Rose CBS Birgit Nilsson canta operas alemanas Nilsson, Birgit London Bolero (Ravel) New York Philharmonic. Dir: Leonard Bernstein Columbia Brahms Double Concerto Perlman-Rostropovich EMI Callas Canta arias de Verdi Callas, María Odeón Callas in París Callas, María EMI Callas sings Verdi at La Scala Callas, María EMI Canta a Italia Schipa, Tito RCA Canzonettas Napolitanas di Steffano, Giuseppe EMI Chopin-Schumann-Rachmaninoff-Liszt Horowitz, Vladimir Columbia Cinderella (Prokofiev) Moscow Radio Symphony Orchestra. Dir: Rozhdestvensky Melodiya Claro de luna Liberace Columbia Coloratura arias Moffo, Anna

92. Boosey And Hawkes: The Home Of Contemporary Music with Pedrell, who aroused his interest in catalan folk music http://www.boosey.com/pages/cr/composer/composer_main.asp?composerid=2716

93. MYNET - Yýllýk 2002 Kripto sistemleri uzmani Dr. Preda Mihailescu, Fransiz matematikçi eugene CharlesCatalan'in (18141894) 1844 yilinda ortaya attigi, hiçbir sayinin http://haber.mynet.com/almanak/detay.asp?lid=6&cid=1

94. Rare Books Quick Links: ISU Library Contains books, letters, manuscripts, articles, pamphlets, clippings, scrapbooks,and similar material related to eugene V. Debs, socialism, and other isms. http://cml.indstate.edu/rare/rare2.html

This page has moved to a new location. Click HERE if your browser doesn't take you there automatically. March 16, 2001

95. Hispanic Translation History, 15th Century In the same period he renders the Decameron into catalan. 1430c. 1431 PopeEugene IV introduces humanist studies at the University of Rome. http://www.fut.es/~apym/on-line/chronology/15.html

ATTEMPT AT A CHRONOLOGY OF HISPANIC TRANSLATION HISTORY FIFTEENTH CENTURY 1400, 5 September. Leonardo Bruni uses traducere and traductio, perhaps for the first time, in a private letter (Folena 1991: 71). 1408 Enrique de Villena is in Barcelona and is somehow associated with the 'Consistorio de la Gaya Ciencia'. 1410-11 Translation into Judeo-Portuguese of the Libro conplido, in an Aljamiad text written in Hebrew characters. call 1412 Catherine of Navarra proclaims that all Jews are to live in separate quarters. 1412-14 Public disputation of Tortosa, held before Pope Benoit XIII and in the presence of about 1000 Christians opposing the rabbis of the Crown of Aragon. The disputation took 22 months to decide whether or not the Messiah had already come, famously resulting in fourteen rabbis converting to Christianity (Marco 1977). . 1415 The manuscript of Ptolemy's Geography is discovered in Florence. Brought there by Paolo Strozzi, a Florentine merchant, it was translated into Latin by Jacopo d'Angelo. 1417 Pope Martin V admits the sovereignty of States by renouncing the Pope's political power over the Councils.

96. ALIQUOT-SEITEN / PRIMZAHLFAMILIEN http://home.t-online.de/home/wolfgang.creyaufmueller/aliquot.htm

Primzahlfamilien - aliquot sequences e-mail tief This page is available in English, too. powered by FreeFind Index ohne frames - Aktivierung der frames mit dem permanenten Inhaltsverzeichnis Definition Inhaltsketten Graphen ... Anfang Definition - Catalan'sche Vermutung hoch tief Eine Kette iterativer Zahleninhalte - kurz Inhaltskette (engl.: aliquot sequence ) - ist eine Folge von natürlichen Zahlen, die mittels der Teilersummenfunktion gebildet werden: s(n) ist die Summe aller natürlichen Teiler einer Zahl n einschließlich der 1 und n selbst. Der Zahleninhalt ist die Teilersumme ohne die Zahl n, die sogenannte ' Summe der echten Teiler i(n) = s(n) - n. In der internationalen Literatur steht für s(n) meist sigma (n). Bei längeren Ketten wird zur Bezeichnung eines bestimmten Gliedes gewöhnlich die erste Zahl geschrieben gefolgt von der Indexnummer gleichbedeutend mit "Inhaltsnummer". Ein Doppelpunkt dient als Trennzeichen. Beispiel: 276:i4 ist der vierte iterative Inhalt der Kette, die mit 276 beginnt, also 276:i4 = 1872. Die ersten Glieder lauten 276, 396, 696, 1104, 1872. Die gesamte Inhaltskette kann eingesehen werden: , ihr Graph ebenso (die Beschreibung der graphischen Darstellung erfolgt weiter unten).

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