21. 6 ABALLAGH M. L’influence des écrits mathématiquesd’ibn albanna sur lesmathématiciens égyptiens de l’empire ottoman, Symposium sur Science, Technology and http://www.ashm.ass.dz/cahier8f/collo8f.htm

6 . Colloques 6. 1. XXe Congrés International d’histoire des Sciences (Liège, Belgique, 20-26/12/1997). Les communications suivantes liées à l’histoire des mathématiques arabes ont été présentés: ABALLAGH M. : L’influence des écrits mathématiquesd’Ibn al-Banna sur les mathématiciens égyptiens de l’empire ottoman, Symposium sur Science, Technology and Industry in the Ottoman World (SU7). (Org.) A. Djebbar et B. Ihsanoglu. ANSARI, S.M.R: The Mathematicien Family of Ahmad Mamar and their Works, Special session of Section 3: Islamic science and technology in Arabic-speaking countries, Central Asia and India, Org.S.M.P. Ansari. BAGHERI, M. : Discovery of a New Letterof al-Kashi, Symposium Science and Technology in anciens and medieval Iran, (Org.) J.P. Hogendijk et M. Bagheri. BELLOSTA, A.: Quelques lectures arabes des Données d’Euclide. Symposium sur The East and the West (S.M.19), Org.: A. Allard, R. Rashed et C. Sasaki. BEN MILED, A.M.: Le Livre X des Eléments d’Euclide dans la tradition arabe, Symposium sur The East and the West (S.M.), Org.: A. Allard, R. Rashed et C. Sasaki. BERGGREN, J.L.: Minor Geometrical Works of al-Kuhi: a Historical and Mathematical Survey, Symposium Science and Technology in ancient and medieval Iran, (Org.) J.P. Hogendijk and M. Bagheri.

22. History Of Astronomy: Persons (I) ibn albanna, al-Marrakushi (1256-1321). Short biography, references andlinks (MacTutor Hist. Math.) Ibn Haiyan, Jabir (?-803) http://www.astro.uni-bonn.de/~pbrosche/persons/pers_i.html

History of Astronomy Persons History of Astronomy: Persons (I) Ibn Al-Banna, al-Marrakushi (1256-1321) Short biography, references and links (MacTutor Hist. Math.) Ibn Haiyan, Jabir (?-803) Short biography (Or see here or here Ibn Qurra: see Thabit Ibn Qurra (836-901) Ibn Qurra: see Ibrahim, ibn Sinan ibn Thabit ibn Qurra (908-946) Ibn Rushd [Averroes; Abul Walid Mahommed Ibn Achmed, Ibn Mahommed Ibn Roschd] (1128-1198) Short biography and references From the Catholic Encyclopedia, 1913 Short biography (Or see here or here Ibn Sina [Avicenna] (980-1037) Short biography and references (MacTutor Hist. Math.) Short biography and references From the Catholic Encyclopedia, 1913 Short biography (Or see here or here Very short biography (Eric Weisstein's Treasure Trove) Crater Avicenna (lunar feature) Ibrahim, ibn Sinan ibn Thabit ibn Qurra (908-946) Short biography and references (MacTutor Hist. Math.) Ingalls, Albert (1888-1958) Very short biography (Eric Weisstein's Treasure Trove) Inghirami, Giovanni (1779-1851)

23. History Of Astronomy: Persons (A) AlBanna, al-Marrakushi ibn see ibn al-banna, al-Marrakushi (1256-1321);Albategnius see al-Battani, Abu Abdallah (c.868-929); al-Battani, http://www.astro.uni-bonn.de/~pbrosche/persons/pers_a.html

History of Astronomy Persons History of Astronomy: Persons (A) Deutsche Fassung Aaronson, Marc Abbadie, Antoine ... Abbon de Fleury [Abbo of Fleury; Abbo von Fleury; Albo; Albon Floriacensis] (c. 945-1004) Abbot, Charles Greeley Abbott, Francis Abbott, Francis (Jnr) Abel, Niels Henrik ... Abiosi [Abbiosi], Giovanni Battista [Jean-Baptiste] (fl. 1490-1520) Ablufarabius: see al-Farabi, Mohammed (c. 870-950) Abraham bar Hiyya Ha-Nasi [Abraham Ben Chaja [Chija]; Abraham Judaeus] (c.1065-c.1136) Short biography and references (MacTutor Hist. Math.) Short biography and references (C. Plicht) Short biography (Encycl. Brit.) Abraham Ben Dior [Ben David, Harischon; Josophat Ben Levi] (12th c.) Very short biography and references (C. Plicht) Abraham ben Meir ibn Ezra: see Ezra, Abraham ben Meir ibn (1092-1167) Abraham Zachut [Abraham Zacuto; Zagut] (c. 1450 - c. 1510) Very short biography and reference (C. Plicht) Abu Abdallah al-Battani (868-929): see al-Battani Abu al-Hasan: see Ali Ibn Rabban al-Tabari (838-870) Abu Ali Hasan Ibn al-Haitham (965-1040): see al-Haitham Abu al-Nasr al-Farabi (870-950): see al-Farabi Abu Dschaasar Almansur (712-775) Short biography and references (C. Plicht)

24. Max Planck Society - EDoc Server ibn al-banna al-Murrakushi , Khazini, Abd al-Rahmanal-K. Authors Abattouy, M. Document type InBook Language English http://edoc.mpg.de/223536

Home News About Us Contact ... Home Search Quick Search Advanced Fulltext Browse Collections Persons My eDoc Session History Login Name: Password: Documentation Help Display Documents ID: , MPI f¼r Wissenschaftsgeschichte / Publications MPIWG [Entries] 'Ibn al-Banna al-Murrakushi', 'Khazini, Abd al-Rahman al-K.' Authors: Abattouy, M. Document type: InBook Language: English Place of Publication: Heidelberg [u.a.] Title of Book: Lexikon der bedeutenden Naturwissenschaftler in drei B¤nden. Bd. 2: F bis Mei Publisher: Spektrum Akademischer Verlag Full Name of Book-Editor(s): Hoffmann, Dieter ; Laitko, Hubert ; M¼ller-Wille, Staffan External Publication Status: published Audience: Not Specified Date of Publication (YYYY-MM-DD): Affiliations: MPI f¼r Wissenschaftsgeschichte Abt. I (Renn) The scope and number of records on eDoc is subject to the collection policies defined by each institute - see "info" button in the collection browse view.

25. Mathematics A commentary on his treatise on arithmetic, written by ibn albanna, gained muchpopularity and was published in French by A. Narre in 1864 and reprinted in http://www.netmuslims.com/info/mathematics.html

Mathematics Muslims have made immense contributions to almost all branches of the sciences and arts, but mathematics was their favourite subject and its development owes a great deal to the genius of Arab and persian scholars. The advancement in different branches of mathematical science commenced during the Caliphate of Omayyads, and Hajjaj bin Yusuf, who was himself a translator of Euclid as well as a great patron of mathematicians. Translations Arithmetic Arabs were the founders of every day arithmetic and taught the use of ciphers to the world. Musa al-Khwarizmi (780850 A.D.) a native of Khwarizm, who lived in the reign of Mamun-ar-Rashid, was one of the greatest mathematicians of all times. He composed the oldest Islamic works on arithmetic and algebra which were the principal source of knowledge on the subject for a fairly long time. George Sarton pays glowing tribute to this outstanding Muslim mathematician and considers him "one of the greatest scientists of his race and the greatest of his time".' He systematised Greek and Hindu mathematical knowledge and profoundly influenced mathematical thought during mediaeval times. He championed the use of Hindu numerals and has the distinction of being the author of the oldest Arabic work on arithmetic known as Kitab-ul Jama wat Tafriq. The original version of this work has disappeared but its Latin translation Trattati a" Arithmetic edited by Bon Compagni in 1157 at Rome is still in existence.

26. New Page 1 for instance, by ibn albanna, 612 in the Kitab Raf al-hijab. of the rulesof calculation.617 ibn al-banna later wrote a commentary on it in a book http://www.muslimphilosophy.com/ik/Muqaddimah/Chapter6/Ch_6_19.htm

The sciences concerned with numbers. The first of them is arithmetic. Arithmetic is the knowledge of the properties of numbers combined in arithmetic or geometric progressions. For instance, in an arithmetic progression, in which each number is always higher by one than the preceding number, the sum of the first and last numbers of the progression is equal to the sum of any two numbers (in the progression) that are equally far removed from the first and the last number, respectively, of the progression. Or, (the sum of the first and last numbers of a progression) is twice the middle number of the progression, if the total number of numbers (in the progression) is an odd number. It can be a progression of even and odd numbers, or of even numbers, or of odd numbers. Or, if the numbers of a (geometrical) progression are such that the first is one-half of the second and the second one­half of the third, and so on, or if the first is one-third of the second and the second one-third of the third, and so on, the result of multiplying the first number by the last number of the progression is equal to the result of multiplying any two numbers of the progression that are equally far removed from the first and the last number, respectively, (of the progression). Or, (the result of multiplying the first number by the last number of a geometrical progression,) if the number of numbers (in the progression) is odd, is equal to the square of the middle number of the progression. For instance, the progression may consist of the powers of two: two, four, eight, sixteen.

27. New Page 1 ibn albanna, 685 wrote an abridgment (of Ibn Ishaq s Zij) which he entitledal-Minhaj. People have been very eager to use the Minhaj, http://www.muslimphilosophy.com/ik/Muqaddimah/Chapter6/Ch_6_21.htm

Astronomy. This science studies the motions of the fixed stars and the planets. From the manner in which these motions take place, astronomy deduces by geometrical methods the existence of certain shapes and positions of the spheres requiring the occurrence of those motions which can be perceived by the senses. Astronomy thus proves, for instance, by the existence of the precession of the equinoxes, that the center of the earth is not identical with the center of the sphere of the sun. Furthermore, from the retrograde and direct motions of the stars, astronomy deduces the existence of small spheres (epicycles) carrying the (stars) and moving inside their great spheres. Through the motion of the fixed stars, astronomy then proves the existence of the eighth sphere. It also proves that a single star has a number of spheres, from the (observation) that it has a number of declinations, and similar things. Only astronomical observation can show the existing motions and how they take place, and their various types. It is only by this means that we know the precession of the equinoxes and the order of the spheres in their different layers as well as the retrograde and direct motions (of the stars), and similar things. The Greeks occupied themselves very much with astronomical observation. They used instruments that were invented for the observation of the motion of a given star. They called them astrolabes. The technique and theory of how to make them, so that their motion conforms to the motion of the sphere, are a (living) tradition among the people.

28. History And Civilization Abd Al-Basit, Ibn Khalil Al-Malati, 1440 Abu Ali ibn albanna, 1005-1100 Autography diary of an eleventh-century historianof Baghdad by George Makdisi, 1958. (41 DS51.B3A2) http://pkukmweb.ukm.my/~library/histciva.htm

History and Civilization 'Abd al-Basit, ibn Khalil al-Malati, 1440-1514 Deux recits de voyage inedits en Afrique du Nord au XVe siecle. Paris : Larose, 1936. 'Abd al-Latif, 1160-1231 Relation de l'Egypte. Paris : Imprimerie Imperiale, 1810. 'Abd Aziz, Muhammad Japan's colonialism and Indonesia. 's-Gravenhage : M. Nijhoff, 1955. Abel, Armand La citadelle Eyyubite de Bosra Eski Cham. Damas, 1956. Abel, Armand Les Musulmans noirs du Maniema, Bruzelles, Publications du centre pour l'etude des problemes du monde Musulman contemporain, 1959. Abu Ali Ibn al-Banna, 1005-1100 Autography diary of an eleventh-century historian of Baghdad by George Makdisi, 1958. Abu Makramah, 1465-1540 Political history of the Yemen at the beginning of the 16th century : Abu Makramah's account of the years 906-927H. (1500-1501 A. D.) with annotations by Lein Oebele Schuman... Abu Nu'aym Ahmad ibn 'Abd Allah, 948-1038 Gschichte Isbahans : nach der leidener handscrift herausgegeben von Sven Dedering. Leiden : E. J. Brill, 1934. Adam, L.

29. List Of Mathematicians: Information From Answers.com alMarrakushi ibn al-banna (Morocco, 1256 - 1321); Abu Arrayhan Muhammad ibnAhmad al-Biruni (Uzbekistan, 973 - 1048); Giacomo Albanese (Italy, Brazil) http://www.answers.com/topic/list-of-mathematicians

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping List of mathematicians Wikipedia @import url(http://content.answers.com/main/content/wp/css/common.css); @import url(http://content.answers.com/main/content/wp/css/gnwp.css); List of mathematicians The famous mathematicians are listed below in English alphabetical transliteration order (by surname Contents: top A B C ... Z A Abu'l-Hasan al-Uqlidisi Abu'l-Wafa (Iran, Niels Henrik Abel (Norway, Shreeram Shankar Abhyankar Abraham bar Hiyya Ha-Nasi Ralph H. Abraham (USA, University of California, Santa Cruz Wilhelm Ackermann (Germany, John Couch Adams (United Kingdom, Robert Adrain (Ireland) Maria Gaetana Agnesi (Italy, Lars Valerian Ahlfors (Finland, Ahmes (Egypt, roughly around 17th century BC Selman Akbulut (Turkey) Yousef Alavi al-Marrakushi ibn Al-Banna (Morocco, Abu Arrayhan Muhammad ibn Ahmad al-Biruni (Uzbekistan, Giacomo Albanese (Italy, Brazil) Jean le Rond d'Alembert (France, Aleksandr Danilovich Aleksandrov (Russia

30. Les Promoteurs De L'esprit Scientifique Dans La Civilisation Islamique ibn al-banna», http://www.isesco.org.ma/pub/FR/Promoteurs/p38.htm

Index 37. Ibn al-Banna «Un savant de Marrakech ferré en plusieurs sciences, qui s'est tout particulièrement distingué en mathématiques, astronomie, astrologie, sciences occultes, mais aussi en médecine».(245) Il s'agit de Ahmed ibn Mohamed ibn Othman al-Azdi, connu comme Abu al-Abbas, Ibn al-Banna al-Marrakchi, par référence à son père qui était maçon (banna). Né à Marrakech en 654 H/1256 où il y a passé la plus grande partie de sa vie, d'où le rattachement de son nom à celui de sa ville natale.(246) C'est là qu'il étudia la grammaire, le hadith et le fiqh (jurisprudence). Il s'est ensuite rendu à Fès pour étudier la médecine, l'astronomie et les mathématiques auprès d'Ibn Makhlouf al-Sijilmassi l'astronome, et Ibn Hijla le mathématicien.(247) Ibn al-Banna a su mériter l'estime des rois Mérinides qui l'invitaient souvent à Fès. Il est mort à Marrakech en 721 H/1321. Contributions scientifiques En matière de calcul, Ibn al-Banna a contribué à l'explication de théories épineuses et de règles inextricables. Il a entrepris des recherches exhaustives sur les fractions et établi des règles pour l'addition des carrés et des cubes, de même que la règle de la double erreur pour la solution des équations du premier degré et des opérations arithmétiques. Il a apporté aussi quelques modifications, sous forme de règle, à la méthode connue comme «la méthode de la simple erreur».(248)

31. Les Promoteurs De L'esprit Scientifique Dans La Civilisation Islamique et Mohamed Ablagh Vie et œuvres d ibn al-banna al-Marrakchi. http://www.isesco.org.ma/pub/FR/Promoteurs/p42.htm

Index Références 1. Ahmed Abdel Baqi : Caractéristiques de la civilisation arabe au troisième siècle de l'Hégire. Centre d'Etudes de l'Unité Arabe, Série «Al-Turath al-Qawmi», 1991. 3. Ibn Abi Usaybaa : Kitab Uyun al-Inbaa. Authentifié par Amer al-Najjar, 1996. 4. Arnold, Sir Thomas : Héritage de l'Islam. Version arabe de Jerses Fathallah, Beyrouth, Dar At-Tali'a, 1972. 5. Ibn Khallikan, Abu Bakr : Wafayatu al-A'yan (Décès des notables). Authentifié par Ihssan Abbas, Beyrouth, Dar al-Sader. 6. Ibn Roshd : Al-Kulliyate fil Tibb. Authentifié par Saïd Shiban et Ammar al-Talbi, Conseil supérieur algérien de la Culture, 1989, Alger. 7. Ibn Zuhr, Abu Marwane Abdul-Malek : Kitab al-Taysir fil al-Mudawat wal Tadbir. Authentifié par Mohamed Ben Abdallah Roudani. Publications de l'Académie du Royaume du Maroc. Série «Al-Turath», 1991. 8. Ibn Tofaïl : Nususs wa Dirassate. Compilation et réimpression de Fouad Sarkis. Institut de l'Histoire des Sciences arabes et islamiques de l'Université de Francfort, Allemagne.

32. L'algèbre Depuis Al-Khwarizmi Jusqu'à Descartes, Par Olivier THILL. ibn al-banna. http://members.aol.com/OlivThill/algebra.htm

L'algèbre dans les années 830 (al-Khwarizmi) à 1637 (Descartes) présenté par Olivier Thill Introduction J'ai écrit cette page pour répondre aux questions suivantes : Qu'est-ce que l'algèbre ? D'où vient ce mot ? Quels furent les développements de l'algèbre chez les musulmans ? Comment l'algèbre fut-il adopté par les chrétiens ? Pourquoi Descartes s'est-il servi de l'algèbre pour résoudre des problèmes de géométrie ? Quelle est l'originalité de Descartes dans le domaine des mathématiques ? Table des paragraphes Introduction 1. Origine du mot Algèbre 2. Contenu du livre ... Conclusion et résumé 1. Origine du mot Algèbre Le mot algèbre vient du titre d'un livre, al-jabr wa'l muqabalah écrit par al-Khwarizmi , vers 830 (prononcer "al-jaber-oual-mouquabalaa" et "al-kouarizmi") Ce livre est dédié au calife al-Mamoun qui régne à Bagdad de 813 à 833 La traduction habituelle du mot jabr est restauration , et dans ce cas précis, il indique le passage d'un terme d'une équation de l'autre côté du signe égal. Par exemple, on fait une "jabr" quand on transforme y + 4 = x en y = x - 4 Le mot muqabalah est traduit par confrontation ou réduction , et dans ce cas précis, il désigne l'opération consistant à éliminer les termes identiques et opposés dans une équation. Par exemple, on fait une "muqabalah", quand on transforme

33. L'algèbre Arabe Translate this page Le livre d’algèbre d’ibn al-banna - Une démonstration algébrique chez ibn al-banna.L’algèbre après le XIIIe siècle - L’algèbre dans les livres de calcul http://www.adapt.snes.edu/article.php3?id_article=78

34. ALHAMBRA 2000 - THE CONGRESS Ibn Sinän s (909946) treatise on The method of analysis and synthesis and otherways of through ibn al-banna writings and those by his commentators, http://www.ugr.es/~alhambra2000/0Congr.htm

ALHAMBRA 2000 European-Arabic Congress of Mathematics (with History of European and Arabic Mathematics and Mathematicians) SCIENTIFIC COMMITTEE President: Karine Chemla (CNRS - U. Paris VII - France) Enrico Giusti (USF - Firenze - Italy) Hourya Sinaceur (CNRS - Paris I - France) CONTENTS AND SCOPE The ALHAMBRA 2000 European-Arabic Congress of Mathematics (with History of European and Arabic Mathematics and Mathematicians) aims for presenting a global view of how mathematical schools around the Mediterranean Sea contributed to the development of Mathematics. It will especially focus on the following points: To offer a synthesis of our recent knowledge on Arabic Mathematics and its transmission towards the North from the 12th till the 17th century onwards, with a special emphasis on the case of Spain. To revise our understanding of the originality of the Mathematics which developed in Europe from the 13th century onwards. The role played by Spain during the Middle Ages in the advancement of Mathematics was twofold. First, the very development of Mathematics during the age of Al-Andalus was very important. On the other hand, various communities lived in Spain in good terms and worked together. Fm the 12th century onwards, this attracted scholars from the North of Europe who came to Spain to translate this body of knowledge into Latin. As a consequence, this symposium is designed for a wide audience, which is not limited to specialists on past and present-day Mathematics, Arabic, Medieval, Renaissance and Classical mathematics.

35. Ethnomathematics Digital Library (EDL) Other terms Gehimab, Ibn’Arabi, Ibn Rushd (Averroe), Ibn Khaldun, ibn albanna,Madinat al-‘ilm, Beit al-Hikma (house of wisdom), textiles, wool, leather, http://www.ethnomath.org/search/browseResources.asp?type=country&id=291

36. Curso Manual Tutorial Translate this page Nombre Al-Marrakushi ibn al-banna. Formato del Recurso Página Web Valoración -De Navegación Buena - De Contenido 9 de 10 http://www.conocimientosweb.net/dcmt/topic1-86.html

Contenidos Inicio Miembros Registro Secciones Editoriales Cursos y manuales Directorio de Cursos, Manuales y Tutoriales Libros digitales gratis ... Trucos Informáticos Servicios Foros de Discusión Descarga Software Enlaces de Interes Enciclopedias y Diccionarios ... Información Patrocinadores Patrocinadores Espacio reservado para patrocinadores, úsalo y prueba la diferencia. Diariotec DVD Jon hackea el Windows Media Player Intel presenta respuesta a demanda de AMD 10 virus más detectados por Virus Radar en Agosto Google Talk! a fondo ... Leer más... Conectados Actualmente hay 159 usuarios. Curso Manual Tutorial: Ficha Técnica Buscar en este Tópico: Ir al Home Selecciona un nuevo Tópico :: Producción de cauho Nombre: Producción de cauho Formato del Recurso: Página Web Valoración: - De Navegación: Buena - De Contenido: 9 de 10 Acceder al recurso editorial Jueves, 16 de Junio del 2005 (20:25:55) (24 Lecturas) Ficha Completa... :: Caucho Natural Nombre: Caucho Natural Formato del Recurso: Página Web Valoración: - De Navegación: Buena - De Contenido: 9 de 10 Acceder al recurso editorial Jueves, 16 de Junio del 2005 (20:22:18) (40 Lecturas)

37. Biography-center - Letter I I llien, Mario www.grandprix.com/gpe/crefillmar.html; ibn al-banna,www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Bann a.html; ibn Sina, http://www.biography-center.com/i.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 I 55 biographies I llien, Mario www.grandprix.com/gpe/cref-illmar.html ibn al-Banna, www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Bann a.html ibn Sina, www-history.mcs.st-and.ac.uk/~history/Mathematicians/Avicenna.ht ml ibn Tahir, www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdad i.html ibn Tibbon, Jacob www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tibb on.html ibn Ishaq Hunayn, www-history.mcs.st-and.ac.uk/~history/Mathematicians/Hunayn.ht ml Ibn Taymiyya, Taqi al-Din geocities.com/pwhce/taymiyyah ibn Yusuf Ahmed, www-history.mcs.st-and.ac.uk/~history/Mathematicians/Ahmed.html Ibrahim, ibn Sinan www-history.mcs.st-and.ac.uk/~history/Mathematicians/Ibrahim.html Ibrahim, Murad Jussuf Bey

38. Islamic Mathematics ibn albanna (Maghreb, ca. 1300). *113. ibn al-banna , Ahmad b. Muhammad.Talkhis a`mal al-hisab. Edited by M. Souissi. Tunis Université de Tunis, 1969. http://www.math.uu.nl/people/hogend/Islamath.html

Bibliography of Mathematics in Medieval Islamic Civilization Version 13 January 1999. This bibliography is a revised, enlarged and updated version of the bibliography on Islamic mathematics by Richard Lorch on pp. 65-86 of Joseph W. Dauben's The History of Mathematics from Antiquity to the Present: A Selective Bibliography , New York and London: Garland, 1985. This bibliography of Islamic mathematics will appear as a chapter in the updated (1999?) version of Dauben's book which will be made available as a CD-Rom. Reactions and suggestions are very welcome, and can be sent to hogend@math.uu.nl . In this preliminary form, no attention has been paid to diacritical marks in Arabic names. The items in the bibliography have been numbered *1, *2, ... *122, *122a, *122b, *123 etc. and many cross-references have been provided. General Introduction Introductory Works Bibliographies and Handbooks Illustrated Works ... Texts and Commentaries (Specific Authors in Chronological Order) Studies on Specific Subjects Transmission of Mathematics Mathematics in Specific Areas in the Islamic World Arithmetic Irrational Magnitudes ... Number Theory, Indeterminate Equations and Magic Squares

39. Ancient Greeks: Prime Numbers And Number Theory For a long period of time, this was the only known pair, until the Arabicmathematician ibn albanna found the next pair 17296 and 18416. http://www.mlahanas.de/Greeks/Primes.htm

Ancient Greeks: Prime Numbers and Number Theory Michael Lahanas Griechische Mathematik: Zahlentheorie und Primzahlen Pythagoras of Samos ( Πυθαγόρας ο Σάμιος) discovered the relation between harmony and numbers. The Pythagoreans saw the number one as the primordial unity from which all else is created. Two was the symbol for the female, three for the male and therefore five (two + three) symbolized marriage. The number four was symbolic of harmony, because two is even, so four (two times two) is "evenly even". Four symbolized the four elements out of which everything in the universe was made (earth, air, fire, and water). Ten that was the sum from one to four was a very special number. The ancient Greeks believed that all numbers had to be rational numbers. 2500 years ago Greeks discovered that if all the common prime numbers were removed from the top and bottom of the ratio then one of the two numbers had to be odd. This we can term reduced form . Obviously, if top and bottom were both even, then both could be divide by the number two and this could be eliminated from both.

40. Deficient Number -- From MathWorld Souissi, M. Un Texte Manuscrit d ibn albanna Al-Marrakusi sur les NombresParfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan Hamdard Nat. http://mathworld.wolfram.com/DeficientNumber.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Special Numbers Digit-Related Numbers Deficient Number Numbers which are not perfect and for which or equivalently where is the divisor function . Deficient numbers are sometimes called defective numbers (Singh 1997). Primes prime powers , and any divisors of a perfect or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, ... (Sloane's SEE ALSO: Abundant Number Least Deficient Number Perfect Number [Pages Linking Here] REFERENCES: Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 3-33, 2005. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 45, 1994. Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 11, 1997. Sloane, N. J. A. Sequences

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