21. Science News Online, Ivars Peterson's MathTrek (12/6/97): The Amazing ABC Conjec That conjecture dates back to 1955, when it was published in Japaneseas a research problem by the late yutaka taniyama. Goro Shimura http://www.sciencenews.org/sn_arc97/12_6_97/mathland.htm

Recently on MathTrek: Getting Slammed in Tennis 11/29/97 Circles of Dissonance 11/22/97 A Progression of Primes 11/15/97 December The Amazing ABC Conjecture In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer. Fermat’s last theorem, for instance, involves an equation of the form x n y n z n . More than 300 years ago, Pierre de Fermat (1601-1665) conjectured that the equation has no solution if x y , and z are all positive integers and n is a whole number greater than 2. Andrew J. Wiles of Princeton University finally proved Fermat’s conjecture in 1994. In order to prove the theorem, Wiles had to draw on and extend several ideas at the core of modern mathematics. In particular, he tackled the Shimura-Taniyama-Weil conjecture, which provides links between the branches of mathematics known as algebraic geometry and complex analysis. That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama. Goro Shimura of Princeton and Andre Weil of the Institute for Advanced Study provided key insights in formulating the conjecture, which proposes a special kind of equivalence between the mathematics of objects called elliptic curves and the mathematics of certain motions in space. The equation of Fermat’s last theorem is one example of a type known as a Diophantine equation an algebraic expression of several variables whose solutions must to be rational numbers (either whole numbers or fractions, which are ratios of whole numbers). These equations are named for the mathematician Diophantus of Alexandria, who discussed such problems in his book

22. Math Forum: Alejandre: Mathematician/Scientist Links Emmy Nunez, Pedro Raman, Chandrasekhra Venkata Ramanujan, Srinivasa Russell, BertrandShihChieh, Chu Somerville, Mary Fairfax taniyama, yutaka Turing, Alan http://mathforum.org/alejandre/workshops/mathematicians.html

Mathematicians/Scientists The names below are possible candidates for research for the second quarter interdisciplinary project for Team 8-1. Select one from the specific list or look at the general list and find one of your own. Read about the person and note: full name date of birth place of birth where educated contribution(s) to mathematics and/or science date of death how studying this person has benefited your life Specific List Agnesi, Maria al'Khwarizmi, Abu Ja'far Muhammad ibn Musa Banneker, Benjamin Bolyai, Janos ... Hypatia again Just, Ernest Khayyam, Omar Kovalevsky, Sonya Kowa, Seki ... Lovelace, Ada Byron again Morgan, Garrett Noether, Emmy Nunez, Pedro Raman, Chandrasekhra Venkata ... Young, Grace Chisholm General List 4000 Years of Women in Science - Time Ordered List African Americans in the Sciences Alphabetical Index of Women Mathematicians Biographies of African-Americans from Maryland ... Search http://mathforum.org/ Please mail comments and suggestions to Suzanne Alejandre

23. Re: Shimura-Taniyama Conjecture By Antreas P. Hatzipolakis of a specific matter. yutaka taniyama (1927 1958) Pleasetell the source of this taniyama quote. (There seem to be so http://mathforum.org/epigone/math-history-list/glexzhangdwimp/v01540B06630B9D985

Re: Shimura-Taniyama Conjecture by Antreas P. Hatzipolakis reply to this message post a message on a new topic Back to messages on this topic Back to math-history-list Subject: Re: Shimura-Taniyama Conjecture Author: xpolakis@otenet.gr Date: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taniyama.html As for the second: The only one reference the authors of the biography above, have in the "References for Yutaka Taniyama" is: G Shimura: Yutaka Taniyama and his Time. Very Personal Recollections. Bull. London Math. Soc. 21 (1989) 186-196. So, most likely the original source is Shimura's article. Antreas The Math Forum

24. Ivars Peterson's MathTrek -Curving Beyond Fermat In the 1950s, Japanese mathematician yutaka taniyama (19271958) proposed that everyrational elliptic curve is a disguised version of a complicated, impossible http://www.maa.org/mathland/mathtrek_11_22_99.html

Ivars Peterson's MathTrek November 22, 1999 Curving Beyond Fermat When Andrew Wiles of Princeton University proved Fermat's last theorem several years ago, he took advantage of recently discovered links between Pierre de Fermat's centuries-old conjecture concerning whole numbers and the theory of so-called elliptic curves. Establishing the validity of Fermat's last theorem involved proving parts of the Taniyama-Shimura conjecture. Four mathematicians have now extended this aspect of Wiles' work, offering a proof of the Taniyama-Shimura conjecture for all elliptic curves rather than just a particular subset of such curves. Mathematicians regard the resulting Taniyama-Shimura theorem as one of the major results of 20th-century mathematics. It establishes a surprising, profound connection between two very different mathematical worlds and, along the way, has important consequences for number theory. An elliptic curve is not an ellipse. It is a solution of a cubic equation in two variables of the form y x ax b (where a and b are fractions, or rational numbers), which can be plotted as a curve made up of one or two pieces.

25. Taniyama-Shimura Theorem - Wikipedia The TaniymaShimura theorem states All elliptic curves over Q are modular. .This theorem was first conjectured by yutaka taniyama in September 1955. http://www.wikipedia.org/wiki/Taniyama-Shimura_conjecture

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles Interlanguage links All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Taniyama-Shimura theorem (Redirected from Taniyama-Shimura conjecture The Taniyama-Shimura theorem establishes an important connection between elliptic curves , which are objects from algebraic geometry , and modular forms , which are certain periodic holomorphic functions investigated in number theory If p is a prime number and E is an elliptic curve over Q , we can reduce the equation defining E modulo p ; for all but finitely many values of p we will get an elliptic curve over the finite field F p , with n p elements, say. One then considers the sequence a p n p p , which is an important invariant of the elliptic curve E . Every modular form also gives rise to a sequence of numbers, by

26. Nanobiographies Translate this page 1923) russe Sylow, Peter Ludwig Mejdell (1832-1918) norvègien Takagi, Teiji (1875-1960)japonais taniyama, yutaka (1027-1958) japonais Tchebychev, Pafnuti http://algo.inria.fr/banderier/Recipro/node53.html

Next: Remerciements Up: Previous: Nanobiographies On devrait trouver ci-dessous les noms des intervenants plus ou moins illustres de ce T.E.R. Bernoulli, Jacques I er (1654-1705) suisse Cassels, John William Scott (1922) anglais Dedekind, Richard Julius Wihelm (1831-1916) allemand Diophante d'Alexandrie (III e Dirichlet, Peter Gustav Lejeune- (1805-1859) allemand Eisenstein, Ferdinand Gotthold (1823-1852) allemand Euclide (III e Euler, Leonhard (1707-1783) allemand Fields, John Charles (1863-1932) canadien Gauss, Karl Friedrich (1777-1855) allemand Goldbach, Hermann (1690-1764) allemand Hardy, Godfrey Harold (1877-1947) anglais Hasse, Helmut (1898-1980) allemand Hecke, Erich (1887-1947) polonais Hensel, Kurt (1861-1941) allemand Hilbert, David (1862-1943) allemand Jacobi, Carl Gustav (1804-1851) allemand Jensen, Johannes Ludwig Wilhelm (1859-1925) allemand Kronecker, Leopold (1823-1891) allemand Kummer, Ernst Eduard (1810-1893) allemand Leibniz, Gottfried William (1646-1716) allemand Linnik, Iurii Vladimirovich (1915-1972) russe Ostrowski, Alexander Markus (1893)

27. Writing Activities Bertrand; ShihChieh, Chu; Somerville, Mary Fairfax; taniyama, yutaka;Turing, Alan; Woods, Granville T. Young, Grace Chisholm. Evaluation http://www.math.wichita.edu/history/activities/writing-act.html

Writing Activity Home Historical Mathematicians Writing Project Grades 5-12 Introduction: Small student groups will use the Internet and other traditional sources to research historically significant mathematicians, write a paper, and creatively present research in class. Prior Knowledge Research skills Paper Writing Skills Word Processing Skills Internet Site Location Skills Student Project Guidelines This in an integrated cooperative learning project for grades 5-12 which allows students to research through Internet sites and traditional methods a historically significant mathematician. Students research the mathematician's personal background and upbringing, education, and their significant contribution to mathematics. Students gain a clearer understanding of a great mathematical discovery and the person behind it. The last component is a creative class presentation/lecture by each group. Learning Advice: As groups work in class, monitor how the groups are working together. I suggest that a teacher schedule 1 or 2 short conferences with the groups to help keep the groups focused and on track. If computer access is limited, I suggest that teachers assign groups a block of time to begin.

28. Sobre La Conjetura De Fermat Translate this page Esta conjetura, hecha por yutaka taniyama (Kisai, 1927 - Tokio, 1958) y por GoroShimura (1930 - ) puede enunciarse de esta manera yutaka taniyama. 06. http://personales.ya.com/casanchi/mat/fermat01.htm

Más artículos de Matemática SOBRE LA CONJETURA DE FERMAT Pierre de Fermat habia escrito que "poseia una demostración maravillosa, que, sin embargo, no cabe en el estrecho márgen de este libro". Parece ser que, o estaba equivocado, o bien hemos perdido un descubrimiento histórico. Trescientos años ha costado dar con una demostración. Se trata de un texto en formato pdf, construido mediante los apartados siguientes: Apartados 1. La Historia. 2. Las infintas soluciones del caso pitagórico. 3. Los intentos de demostración clásicos de la conjetura. 4. Curvas elípticas. Curvas elípticas modulares. 5. La conjetura de Taniyama-Shimura. 6. El trabajo de Gerard Frey y Kenneth Ribes. 7. Andrew Wiles. 8. Documentación. Para obtener el texto completo en formato pdf, pinche aquí: ConjeturaFermat.pdf Más artículos de Matemática

29. Fermat's Last Theorem By J-P Guegan related to Fermat's Last Theorem. yutaka taniyama was conductingresearch into elliptical curves. These curves take the form of http://students.bath.ac.uk/ns0jpg/page6.html

A Modern Breakthrough The late 19th century and the early 20th century was another slow period in the research into Fermat's Last Theorem, due in part to the First World War and other conflicts around the globe. At these times many of the world's greatest minds were focused on inventing new, more effective weapons. During the early nineteen hundreds Fermat's Theorem did gain a rather dubious honour, it became the theory which had the most false proofs presented about it. For example in the four years between 1908 to 1912 over 1000 false proofs were presented. This is not to say that all these proofs were formatted by charlatans, however many of them were due to the substantial prize money on offer. This high figure can also be put down to small gaps or errors in long complicated proofs. There was nonetheless some progress in the field as computers were used to test theories and hence proofs could be found up to n=4,000,000 by 1993. The first major breakthrough for centuries came in 1955, although it was not thought to be related to Fermat's Last Theorem. Yutaka Taniyama was conducting research into elliptical curves. These curves take the form of y2 = x2 + ax + b where a and b are constants and were found to be closely linked to Fermat's Last Theorem. After research by two further scientists, Shimura and Weil, a conjecture, called the Shimura-Taniyama-Weil conjecture was produced. This was found in 1986 to be connected to Fermat's Last Theorem, and shown that, far from being a trivial brainteaser, as Gauss had suggested, Fermat's Last Theorem was closely related to the fundamental properties of space.

30. Who Was Responsible For Solving Fermat's Last Theorem? 1955 yutaka taniyama (19271958) Goro Shimura. taniyama and Shimura helpedorganise the Tokyo-Nikko Symposium on Algebraic Number Theory. http://students.bath.ac.uk/ns0jeb/those.html

So Who Solved Fermat's Last Theorem? Fermat's Last Theorem is such an immensely complex problem, that it would be impossile to say it was solved by the one person. It took over 350 years, and some of the finest mathematicians ever to solve this puzzle. These are the many involved. Leonhard Euler (1707-1783) Euler proved FLT for n = 3 and 4 independently. Euler invented the imaginary number, i, a nontrivial number, equal to -1 , and created a new field, topology. Carl Friedrich Gauss (1777-1855) Gauss published Disquisitiones Arithmeticae. Gauss studied the behaviour of functions on the complex plane. Some of these analytic functions, called modular forms, turned out to be crucial to the new approaches to the Theorem. Sophie Germain (1776-1831) Sophie Germain's theorem states that if a solution of Fermat's equation for n = 5 existed, all three numbers must be divisible by 5. The theorem divides Fermat's last theorem into two cases: Case I for numbers that are not divisible by 5, and Case II for numbers that are. Gabriel Lamé (1795-1870) Lamé proved FLT for n = 7 in 1839. He subsequently suggested a general approach to the problem and factored the left side of Fermat's equation, xn + yn, into linear factors using complex numbers, but the factorisation he suggested was not unique, and therefore there was no solution.

31. Www.math.wisc.edu/~propp/courses/491/articles yutaka taniyama and his time very personal recollections , by Goro Shimura; Bulletinof the London Mathematical Society, Volume 21 (1989), pages 186196. http://www.math.wisc.edu/~propp/courses/491/articles

32. News Media Tip - November 19, 1999 For more than 30 years, the conjecture by yutaka taniyama and Goro Shimura that''every elliptic curve over the rational numbers is 'modular' has http://www.nsf.gov/od/lpa/news/tips/99/tip91119.htm

News Media Tip - November 19, 1999 For more information on these science news and feature story tips, please contact the public information officer at the end of each item at (703) 292-8070. Editor: Cheryl Dybas Contents of this Tipsheet: NEW DATA BASE ON REACTIVE CHLORINE EMISSIONS IN THE ATMOSPHERE SOLUTION TO 40-YEAR-OLD MATHEMATICAL PROBLEM HAILED AS BREAKTHROUGH THE COST OF STORMS New Data Base On Reactive Chlorine Emissions In The Atmosphere An international team of atmospheric chemists has produced the first gridded global inventory of reactive chlorine emissions to the atmosphere. "This work provides an objective benchmark for assessing our understanding of the global chlorine cycle, and for investigating the potential environmental implications of future changes in chlorine emissions," says scientist William Keene of the University of Virginia at Charlottesville, one of 18 investigators working on the project. Individual investigators received support from various sources, including the National Science Foundation. The project is similar to the global inventory of carbon emissions conducted several years ago to investigate natural and human influences on atmospheric carbon concentrations. That study was central to the discussions leading to the recent Kyoto protocols, a set of international guidelines for regulating future carbon emissions.

33. VACETS Technical Column - Tc58 The taniyamaShimura conjecture, introduced by the Japanese yutaka taniyama in1955 and later improved by Goro Shimura of Princeton University, stated that http://www.vacets.org/sfe/fermat.html

VACETS Regular Technical Column "Science for Everyone" "Science for Everyone" was a technical column posted regularly on the VACETS forum. The author of the following articles is Dr. Vo Ta Duc . For more publications produced by other VACETS members, please visit the VACETS Member Publications page or Technical Columns page The VACETS Technical Column is contributed by various members , especially those of the VACETS Technical Affairs Committe. Articles are posted regulary on vacets@peak.org forum. Please send questions, comments and suggestions to vacets-ta@vacets.org Mon, 24 Oct 1994 FERMAT'S LAST THEOREM In the [SCIENCE FOR EVERYONE] column last week, I had three bonus problems posted and no one had solved any of them. All I heard was all kinds of discussion about the first bonus, the Fermat's last theorem. It asserts that "For any integer n greater than 2, the equation (a^n + b^n = c^n) has no solutions for which a, b, and c are integers greater than zero." The discussion was interesting. Actually, I had heard that someone had found a solution to the theorem sometime last year. A few months ago, I heard that the proof had some holes in it; some are small like pin-holes and some are as big as black holes. All the pin-holes, potholes, manholes were filled, but the biggest hole, the black hole, was not filled. I guess that there is no way to fill a black hole. It just swallows everything you throw at it and gets bigger. I didn't pay much attention until last week when I saw that many people were discussing it. I decided to do some research into it, and here is what I found. This story is rather long, so I'm going to present it in an unusual way by summarizing the results first. This is so people who do not have time to follow the whole story, grasp at least grasp some idea.

34. Info Humaniora Onze-Lieve-Vrouwecollege Assebroek Keuze1 JAPAN. In 1960 bestudeert de wiskundige yutaka taniyama (19271958) elliptischekrommen, die gedefinieerd worden door vergelijkingen van de volgende (of een http://users.pandora.be/olva/html/wis/fermat.html

De laatste stelling van FERMAT BABYLONIE Op een kleitafeltje (rond 1950 voor Christus) vinden we in spijkerschrift de eerste strikt positieve gehele oplossingen van de vergelijking x² + y² = z². Eén van de drietallen uit die tijd was: x=120, y=119 en z=169 zodat (120)² + (119)² = (169)². Het populairste drietal was vermoedelijk toen reeds (3,4,5): (3)²+(4)²=(5)². Ook het drietal (12,5,13) was reeds bekend: (12)²+(5)²=(13)². De formule x² + y² = z² is voor de Babyloniërs in de eerste plaats een betrekking tussen drie getallen. Vermoedelijk waren de Babyloniërs in staat oplossingen (x,y,z) te genereren met de volgende methode: Beschouw twee natuurlijke getallen, die relatief priem zijn (en dus geen gemeenschappelijke deler hebben, behalve het getal 1). Neem u als grootste en v als kleinste getal. Stel nu: x = 2 u v en y = u² - v² en z = u² + v². Het drietal (x,y,z) is een oplossing van de vergelijking x² + y² = z². Op het kleitafeltje "PLIMPTON 322' vinden we het drietal (13500,12709,18541). Het wordt gevonden via de waarden u=125 en v=54. Controleer dat (13500)² + (12709)² = (18541)². GRIEKENLAND De vergelijking x² + y² = z² heeft vele strikt positieve gehele oplossingen. Deze oplossingen worden pythagorische drietallen genoemd naar de Griekse filosoof en wiskundige Pythagoras (580-500 voor Christus). Voor de Grieken heeft de formule x² + y² = z² een meetkundige betekenis en de getallen x, y en z verwijzen naar de zijden van een rechthoekige driehoek.

35. Timeline Of Fermat's Last Theorem 1955, yutaka taniyama (19271958) Goro Shimura, taniyama and Shimura helpedorganize the Tokyo-Nikko Symposium on Algebraic Number Theory. http://www.public.iastate.edu/~kchoi/time.htm

Drink to Me (Carolan, sequenced by Barry Taylor) Timeline of Fermat's Last Theorem when who what 1900 BC Babylonians A clay tablet, now in the museum of Columbia University, called Plimpton 322, contains 15 triples of numbers. They show that a square can be written as the sum of two smaller squares, e.g., 5 circa 530 Pythagoras Pythagoras was born in Samos. Later he spent 13 years in Babylon, and probably learned the Babylonian's results, now known as the Pythagorean triples. Pythagoras was also the founder of a secret society that studied among others "perfect" numbers. A perfect number is one that is the sum of its multiplicative factors. For instance, 6 is a perfect number (6 = 1 + 2 + 3). Pythagoreans also recognized that 2 is an irrational number. circa 300 BC Euclid of Alexandria Euclid is best known for his treatise Elements circa 400 BC Eudoxus Eudoxus was born in Cnidos, and became a colleague of Plato. He contributed to the theory of proportions, and invented the "method of exhaustion." This is the same method employed in integral calculus. circa 250 AD Diophantus of Alexandria Diophantus wrote Arithmetica , a collection of 130 problems giving numerical solutions, which included the Diophantine equations , equations which allow only integer solutions (e.g, ax + by = c, x

36. Nature Publishing Group As long ago as 1954, two Japanese mathematicians, yutaka taniyama andGoro Shimura, had suggested that these two areas were connected. http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v387/n6636/full/

37. Retiro Cultural - O Último Teorema De Fermat in 1832. yutaka taniyama, whose insights would ultimately lead tothe solution, tragically killed himself in 1958. On the other http://www.angelfire.com/ab/geloneze/fermat.html

Retiro Cultural - O Último Teorema de Fermat Matematicoterapia 1 Matematicoterapia 2 A história da demonstração da conjectura mais famosa da Matemática Um problema que desafiou os matemáticos por mais de 300 anos Baseado nos livros "O Último Teorema de Fermat" de Simon Singh, edição brasileira pela Editora Record, 1998, e no livro "Fermat’s Last Theorem:Unlocking the Secret of an Ancient Mathematical Problem" By Amir D. Aczel Delta - Trade Paperbacks "This is a captivating volume ... The brilliant backdoor method used by Mr. Wiles as he reached his solution, along with the debt he owed to many other contemporary mathematicians, is graspable in Mr. Aczel’s lucid prose. Equally important is the sense of awe that Mr. Aczel imparts for the hidden, mystical harmonies of numbers, and for that sense of awe alone, his slender volume is well worth the effort." —The New York Times "For more than three centuries, Fermat’s Last Theorem was the most famous unsolved problem in mathematics; here’s the story of how it was solved ... An excellent short history of mathematics, viewed through the lens of one of its great problems — and achievements." — Kirkus Reviews "This exciting recreation of a landmark discovery reveals the great extent to which modern mathematics is a collaborative enterprise ... While avoiding technical details, Aczel maps the strange, beautiful byways of modern mathematical thought in ways the layperson can grasp."

38. Pgsql-jp pgsqljp 13555 Re Movie? yutaka tanida yutaka@marin.or jp13497 Re backend terminated abnormally taniyama Hideki yoko http://ml.postgresql.jp/pgsql-jp-old/pgsql-jp/2000Mar/

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39. Pgsql-jp pgsqljp 7208 Re yutaka Nakamura 27493u pgsql-jp 7135Re semget failed (No space left on device) ? taniyama Hideki yoko http://ml.postgresql.jp/pgsql-jp-old/pgsql-jp/1999Feb/

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40. Zimaths: Fermat's Last Theorem But progress was made, notably by the Japanese mathematicians yutaka taniyama (whokilled himself in 1958) and Goro Shimura (who's a professor at Princeton http://www.uz.ac.zw/science/maths/zimaths/flt.htm

Fermat's Last Theorem by Dino Surendran Mathematians do not often make it into the world's press. But in 1993, Andrew Wiles, a British maths professor at Princeton University, hit the headlines. His feat? Showing that there are no integer solutions to the equation x n + y n = z n when n is an integer greater than 2. In other words, he had proved Fermat's Last Theorem This problem was written down around 1637 by Pierre de Fermat, a French lawyer in Toulouse who was also a prominent amateur mathematician. He was reading a textbook when a thought occurred to him. He decided to write it down before he forgot it - and the nearest piece of paper was the margin of the said textbook: ``On the other hand it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or in general, for any number which is a power greater than the second to be written as a sum of two like powers. I have a truly marvellous demonstration of this proposition which this margin is too large to contain.'' Now, it is suspected that he later found that his proof was incorrect, since he only ever

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