TURING and the PRIMES Alan Turing's Exploits in Code Breaking, Philosophy, Artificial Intelligence and the Founda- Tions of Co
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A Tour of Fermat's World
ATOUR OF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic ATOUR OF FERMAT’S WORLD Congruent numbers Fermat’s infinite descent Counting solutions Ching-Li Chai Zeta functions and their special values Department of Mathematics Modular forms and University of Pennsylvania L-functions Elliptic curves, complex multiplication and Philadelphia, March, 2016 L-functions Weil conjecture and equidistribution ATOUR OF Outline FERMAT’S WORLD Ching-Li Chai 1 Samples of numbers Samples of numbers More samples in 2 More samples in arithemetic arithemetic Congruent numbers Fermat’s infinite 3 Congruent numbers descent Counting solutions 4 Fermat’s infinite descent Zeta functions and their special values 5 Counting solutions Modular forms and L-functions Elliptic curves, 6 Zeta functions and their special values complex multiplication and 7 Modular forms and L-functions L-functions Weil conjecture and equidistribution 8 Elliptic curves, complex multiplication and L-functions 9 Weil conjecture and equidistribution ATOUR OF Some familiar whole numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in §1. Examples of numbers arithemetic Congruent numbers Fermat’s infinite 2, the only even prime number. descent 30, the largest positive integer m such that every positive Counting solutions Zeta functions and integer between 2 and m and relatively prime to m is a their special values prime number. Modular forms and L-functions 3 3 3 3 1729 = 12 + 1 = 10 + 9 , Elliptic curves, complex the taxi cab number. As Ramanujan remarked to Hardy, multiplication and it is the smallest positive integer which can be expressed L-functions Weil conjecture and as a sum of two positive integers in two different ways. -
A Primer of Analytic Number Theory: from Pythagoras to Riemann Jeffrey Stopple Index More Information
Cambridge University Press 0521813093 - A Primer of Analytic Number Theory: From Pythagoras to Riemann Jeffrey Stopple Index More information Index #, number of elements in a set, 101 and (2n), 154–157 A =, Abel summation, 202, 204, 213 and Euler-Maclaurin summation, 220 definition, 201 definition, 149 Abel’s Theorem Bernoulli, Jacob, 146, 150, 152 I, 140, 145, 201, 267, 270, 272 Bessarion, Cardinal, 282 II, 143, 145, 198, 267, 270, 272, 318 binary quadratic forms, 270 absolute convergence definition, 296 applications, 133, 134, 139, 157, 167, 194, equivalence relation ∼, 296 198, 208, 215, 227, 236, 237, 266 reduced, 302 definition, 133 Birch Swinnerton-Dyer conjecture, xi, xii, abundant numbers, 27, 29, 31, 43, 54, 60, 61, 291–294, 326 82, 177, 334, 341, 353 Black Death, 43, 127 definition, 27 Blake, William, 216 Achilles, 125 Boccaccio’s Decameron, 281 aliquot parts, 27 Boethius, 28, 29, 43, 278 aliquot sequences, 335 Bombelli, Raphael, 282 amicable pairs, 32–35, 39, 43, 335 de Bouvelles, Charles, 61 definition, 334 Bradwardine, Thomas, 43, 127 ibn Qurra’s algorithm, 33 in Book of Genesis, 33 C2, twin prime constant, 182 amplitude, 237, 253 Cambyses, Persian emperor, 5 Analytic Class Number Formula, 273, 293, Cardano, Girolamo, 25, 282 311–315 Catalan-Dickson conjecture, 336 analytic continuation, 196 Cataldi, Pietro, 30, 333 Anderson, Laurie, ix cattle problem, 261–263 Apollonius, 261, 278 Chebyshev, Pafnuty, 105, 108 Archimedes, 20, 32, 92, 125, 180, 260, 285 Chinese Remainder Theorem, 259, 266, 307, area, basic properties, 89–91, 95, 137, 138, 308, 317 198, 205, 346 Cicero, 21 Aristotle’s Metaphysics,5,127 class number, see also h arithmetical function, 39 Clay Mathematics Institute, xi ∼, asymptotic, 64 comparison test Athena, 28 infinite series, 133 St. -
A Theorem of Fermat on Congruent Number Curves Lorenz Halbeisen, Norbert Hungerbühler
A Theorem of Fermat on Congruent Number Curves Lorenz Halbeisen, Norbert Hungerbühler To cite this version: Lorenz Halbeisen, Norbert Hungerbühler. A Theorem of Fermat on Congruent Number Curves. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2019, Hardy-Ramanujan Journal, 41, pp.15 – 21. hal-01983260 HAL Id: hal-01983260 https://hal.archives-ouvertes.fr/hal-01983260 Submitted on 16 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Hardy-Ramanujan Journal 41 (2018), 15-21 submitted 28/03/2018, accepted 06/07/2018, revised 06/07/2018 A Theorem of Fermat on Congruent Number Curves Lorenz Halbeisen and Norbert Hungerb¨uhler To the memory of S. Srinivasan Abstract. A positive integer A is called a congruent number if A is the area of a right-angled triangle with three rational sides. Equivalently, A is a congruent number if and only if the congruent number curve y2 = x3 − A2x has a rational point (x; y) 2 Q2 with y =6 0. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order. -
Congruent Numbers with Many Prime Factors
Congruent numbers with many prime factors Ye Tian1 Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China † Edited by S. T. Yau, Harvard University, Cambridge, MA, and approved October 30, 2012 (received for review September 28, 2012) Mohammed Ben Alhocain, in an Arab manuscript of the 10th Remark 2: The kernel A½2 and the image 2A of the multiplication century, stated that the principal object of the theory of rational by 2 on A are characterized by Gauss’ genus theory. Note that the × 2 right triangles is to find a square that when increased or diminished multiplication by 2 induces an isomorphism A½4=A½2’A½2 ∩ 2A. by a certain number, m becomes a square [Dickson LE (1971) History By Gauss’ genus theory, Condition 1 is equivalent in that there are of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In exactly an odd number of spanning subtrees in the graph whose modern language, this object is to find a rational point of infinite ; ⋯; ≠ vertices are p0 pk and whose edges are those pipj, i j,withthe order on the elliptic curve my2 = x3 − x. Heegner constructed such quadratic residue symbol pi = − 1. It is then clear that Theorem 1 rational points in the case that m are primes congruent to 5,7 mod- pj ulo 8 or twice primes congruent to 3 modulo 8 [Monsky P (1990) follows from Theorem 2. Math Z – ’ 204:45 68]. We extend Heegner s result to integers m with The congruent number problem is not only to determine many prime divisors and give a sketch in this report. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Arxiv:0903.4611V1 [Math.NT]
RIGHT TRIANGLES WITH ALGEBRAIC SIDES AND ELLIPTIC CURVES OVER NUMBER FIELDS E. GIRONDO, G. GONZALEZ-DIEZ,´ E. GONZALEZ-JIM´ ENEZ,´ R. STEUDING, J. STEUDING Abstract. Given any positive integer n, we prove the existence of in- finitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point Pλ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X3 n2X over Q(λ). − 1. Congruent numbers over the rationals A positive integer n is called a congruent number if there exists a right ∗ triangle with rational sides and area equal to n, i.e., there exist a, b, c Q with ∈ 2 2 2 1 a + b = c and 2 ab = n. (1) It is easy to decide whether there is a right triangle of given area and inte- gral sides (thanks to Euclid’s characterization of the Pythagorean triples). The case of rational sides, known as the congruent number problem, is not completely understood. Fibonacci showed that 5 is a congruent number 3 20 41 (since one may take a = 2 , b = 3 and c = 6 ). Fermat found that 1, 2 and 3 are not congruent numbers. Hence, there is no perfect square amongst arXiv:0903.4611v1 [math.NT] 26 Mar 2009 the congruent numbers since otherwise the corresponding rational triangle would be similar to one with area equal to 1. -
Who Did What? Engineers Vs Mathematicians Title Page America Vs Europe Ideas Vs Implementations Babbage Lovelace Turing
Who did what? Engineers vs Mathematicians Title Page America vs Europe Ideas vs Implementations Babbage Lovelace Turing Flowers Atanasoff Berry Zuse Chandler Broadhurst Mauchley & Von Neumann Wilkes Eckert Williams Kilburn SlideSlide 1 1 SlideSlide 2 2 A decade of real progress What remained to be done? A number of different pioneers had during the 1930s and 40s Memory! begun to make significant progress in developing automatic calculators (computers). Thanks (almost entirely) to A.M. Turing there was a complete theoretical understanding of the elements necessary to produce a Atanasoff had pioneered the use of electronics in calculation stored program computer – or Universal Turing Machine. Zuse had mad a number of theoretical breakthroughs Not for the first time, ideas had outstripped technology. Eckert and Mauchley (with JvN) had produced ENIAC and No-one had, by the end of the war, developed a practical store or produced the EDVAC report – spreading the word memory. Newman and Flowers had developed the Colossus It should be noted that not too many people were even working on the problem. SlideSlide 3 3 SlideSlide 4 4 The players Controversy USA: Eckert, Mauchley & von Neumann – EDVAC Fall out over the EDVAC report UK: Maurice Wilkes – EDSAC Takeover of the SSEM project by Williams (Kilburn) UK: Newman – SSEM (what was the role of Colossus?) Discord on the ACE project and the departure of Turing UK: Turing - ACE EDSAC alone was harmonious We will concentrate for a little while on the SSEM – the winner! SlideSlide 5 5 SlideSlide 6 6 1 Newman & technology transfer: Secret Technology Transfer? the claim Donald Michie : Cryptanalyst An enormous amount was transferred in an extremely seminal way to the post war developments of computers in Britain. -
Hereby the Screen Stands in For, and Thereby Occludes, the Deeper Workings of the Computer Itself
John Warnock and an IDI graphical display unit, University of Utah, 1968. Courtesy Salt Lake City Deseret News . 24 doi:10.1162/GREY_a_00233 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/GREY_a_00233 by guest on 27 September 2021 The Random-Access Image: Memory and the History of the Computer Screen JACOB GABOURY A memory is a means for displacing in time various events which depend upon the same information. —J. Presper Eckert Jr. 1 When we speak of graphics, we think of images. Be it the windowed interface of a personal computer, the tactile swipe of icons across a mobile device, or the surreal effects of computer-enhanced film and video games—all are graphics. Understandably, then, computer graphics are most often understood as the images displayed on a computer screen. This pairing of the image and the screen is so natural that we rarely theorize the screen as a medium itself, one with a heterogeneous history that develops in parallel with other visual and computa - tional forms. 2 What then, of the screen? To be sure, the computer screen follows in the tradition of the visual frame that delimits, contains, and produces the image. 3 It is also the skin of the interface that allows us to engage with, augment, and relate to technical things. 4 But the computer screen was also a cathode ray tube (CRT) phosphorescing in response to an electron beam, modified by a grid of randomly accessible memory that stores, maps, and transforms thousands of bits in real time. The screen is not simply an enduring technique or evocative metaphor; it is a hardware object whose transformations have shaped the ma - terial conditions of our visual culture. -
Social Construction and the British Computer Industry in the Post-World War II Period
The rhetoric of Americanisation: Social construction and the British computer industry in the Post-World War II period By Robert James Kirkwood Reid Submitted to the University of Glasgow for the degree of Doctor of Philosophy in Economic History Department of Economic and Social History September 2007 1 Abstract This research seeks to understand the process of technological development in the UK and the specific role of a ‘rhetoric of Americanisation’ in that process. The concept of a ‘rhetoric of Americanisation’ will be developed throughout the thesis through a study into the computer industry in the UK in the post-war period . Specifically, the thesis discusses the threat of America, or how actors in the network of innovation within the British computer industry perceived it as a threat and the effect that this perception had on actors operating in the networks of construction in the British computer industry. However, the reaction to this threat was not a simple one. Rather this story is marked by sectional interests and technopolitical machination attempting to capture this rhetoric of ‘threat’ and ‘falling behind’. In this thesis the concept of ‘threat’ and ‘falling behind’, or more simply the ‘rhetoric of Americanisation’, will be explored in detail and the effect this had on the development of the British computer industry. What form did the process of capture and modification by sectional interests within government and industry take and what impact did this have on the British computer industry? In answering these questions, the thesis will first develop a concept of a British culture of computing which acts as the surface of emergence for various ideologies of innovation within the social networks that made up the computer industry in the UK. -
Manchester Baby Simulator Narration
Manchester Baby Simulator Narration By David Sharp The University of Warwick 11th January 2001 Geoff Tootill’s Highest Common Factor program, written in his notebook 19/6/48. Recent background1 Between 1996 and 1998 much research was done by the Manchester Computer Conservation Society in building an accurate replica of the ‘Baby’ (as of 21st June 1948) for its 50th anniversary. A competition was also staged for people to write programs for the ‘Baby’, the winning program was run on the replica on the 21st June 1998. This rebuild of the replica raised the profile of the Baby and also caused a lot of research into details of the machine that might otherwise have been lost to history. Historical accuracy The display presented by the simulator is a stylized version of the original June 1948 Baby. I have made the interface as much like the original as is possible using Java’s GUI capability. Great pains have been taken to provide an accurate rendition of the Baby’s controls and idiosyncrasies. As well as this original interface, I have borrowed from the “player” notion of Martin Campbell-Kelly’s EDSAC simulator (Raul and Hashagen, p. 400) in providing an alternative modern day interface for loading and running the ‘Baby’. The similarities between the original machine and the simulator interface are shown below in Figure 1. It would be easy for someone to learn to operate the original machine from the simulator. Figure 1: The original Baby’s (above), the replica’s (above right) and the simulator’s (right) interfaces. 1 The history and widespread influence of the development of the Baby is outlined in the introduction to the user guide. -
ATLAS@50 Then and Now
ATLAS@50 Then and now Science & Technology Facilities Council The origins of Atlas In the years following the Second World War, there was huge interest in nuclear physics technologies both for civilian nuclear energy research and for military nuclear weapons. These technologies required ever-increasing amounts of computer power to analyse data and, more importantly, to perform simulations far more safely than would be possible in a laboratory. By 1958, 75% of Britain’s R&D computer power was provided by the Atomic Energy Authority (UKAEA) at Aldermaston (using an American IBM 704), Risley, Winfrith and Harwell (each using a British Ferranti Mercury). These were however what we would now call The advisory panel for the super-computer single-user systems, capable of running only included Freddy Williams (a pioneer in radar and Foreword one program at a time, so were essentially very computer technology) from Victoria University large desktop calculators. Indeed, the name of Manchester, Maurice Wilkes (designer of It is now known as Rutherford Appleton Laboratory buildings R26 and R27, but over “computer” referred to the human operator; EDSAC, the Electronic Delay Storage Automatic its 50+ year history, the Atlas Computer Laboratory in the parish of Chilton has been the machinery itself was known as an Calculator) from Cambridge, Albert Uttley of the “automatic calculating machine.” RRE (Royal Radar Establishment), and Ted Cooke- part of many seminal moments in computing history. Yarborough (designer of the Dekatron and CADET British manufacturers of automatic calculating computers) from AERE Harwell; Christopher Over its 50+ year history it has also housed It was cleaned and restored as far as we could, machines were doing well until IBM announced Strachey (a pioneer in programming language many mainframe and super-computers including and took pride of place at an event held to that their first transistorised computer was to design) was in charge of design. -
The Manchester University "Baby" Computer and Its Derivatives, 1948 – 1951”
Expert Report on Proposed Milestone “The Manchester University "Baby" Computer and its Derivatives, 1948 – 1951” Thomas Haigh. University of Wisconsin—Milwaukee & Siegen University March 10, 2021 Version of citation being responded to: The Manchester University "Baby" Computer and its Derivatives, 1948 - 1951 At this site on 21 June 1948 the “Baby” became the first computer to execute a program stored in addressable read-write electronic memory. “Baby” validated the widely used Williams- Kilburn Tube random-access memories and led to the 1949 Manchester Mark I which pioneered index registers. In February 1951, Ferranti Ltd's commercial Mark I became the first electronic computer marketed as a standard product ever delivered to a customer. 1: Evaluation of Citation The final wording is the result of several rounds of back and forth exchange of proposed drafts with the proposers, mediated by Brian Berg. During this process the citation text became, from my viewpoint at least, far more precise and historically reliable. The current version identifies several distinct contributions made by three related machines: the 1948 “Baby” (known officially as the Small Scale Experimental Machine or SSEM), a minimal prototype computer which ran test programs to prove the viability of the Manchester Mark 1, a full‐scale computer completed in 1949 that was fully designed and approved only after the success of the “Baby” and in turn served as a prototype of the Ferranti Mark 1, a commercial refinement of the Manchester Mark 1 of which I believe 9 copies were sold. The 1951 date refers to the delivery of the first of these, to Manchester University as a replacement for its home‐built Mark 1.