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Chen Prime Liczby Pierwsze Chena
Chen Prime Liczby pierwsze Chena Chen Jingrun • Data urodzenia: 22 maj 1933 • Data śmierci: 19 marzec 1996 Pochodzi z wielodzietnej rodziny z Fuzhou, Fujian, Chiny. W 1953 roku skończył wydział matematyki na Uniwersytecie w Xiamen. Jego prace nad przypuszczeniem o bliźniaczych liczbach pierwszych oraz hipotezą Goldbacha doprowadziły do postępu analitycznej teorii liczb. Największym jego osiągnięciem było tzw. twierdzenie Chena stanowiące słabszą wersję słynnej hipotezy Goldbacha. Nazwiskiem Chen Jingruna została nazwana planetoida 7681 Chenjingrun odkryta w 1996 roku Hipoteza Goldbacha • jeden z najstarszych nierozwiązanych problemów w teorii liczb, liczy sobie ponad 250 lat • W 1742 roku, w liście do Leonharda Eulera, Christian Goldbach postawił hipotezę: każda liczba naturalna większa niż 2 może być przedstawiona w postaci sumy trzech liczb pierwszych (ta sama liczba pierwsza może być użyta dwukrotnie) Euler po otrzymaniu listu stwierdził iż hipotezę Goldbacha można uprościć i przedstawić ją w następujący sposób: każda liczba naturalna parzysta większa od 2 jest sumą dwóch liczb pierwszych Powyższą hipotezę do dzisiaj nazywaną "hipotezą Goldbacha" sformułował w rezultacie Euler, jednak nazwa nie została zmieniona. Oto kilka prostych przykładów: 4=2+2 6=3+3 8=3+5 10=3+7=5+5 … 100=53+47… Dzięki użyciu komputerów udało się pokazać, że hipoteza Goldbacha jest prawdziwa dla liczb naturalnych mniejszych niż 4 × 1017 (przez przedstawienie każdej z tych liczb w postaci sumy dwóch liczb pierwszych). Co więcej, większość współczesnych matematyków -
Goldbach, Christian
CHRISTIAN GOLDBACH (March 18, 1690 – November 20, 1764) by HEINZ KLAUS STRICK, Germany One of the most famous, still unproven conjectures of number theory is: • Every even number greater than 2 can be represented as the sum of two prime numbers. The scholar CHRISTIAN GOLDBACH made this simple mathematical statement to his pen pal LEONHARD EULER in 1742 as an assumption. (In the original version it said: Every natural number greater than 2 can be represented as the sum of three prime numbers, since at that time the number 1 was still considered a prime number.) All attempts to prove this theorem have so far failed. Even the offer of a prize of one million dollars hardly led to any progress. CHEN JINGRUN (1933-1996, Chinese stamp on the left), student of HUA LUOGENG (1910-1985, stamp on the right), the most important Chinese mathematician of the 20th century, succeeded in 1966 in making the "best approximation" to GOLDBACH's conjecture. CHEN JINGRUN was able to prove that any sufficiently large even number can be represented as the sum of a prime number and another number that has at most two prime factors. Among the first even numbers are those that have only one GOLDBACH decomposition: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 12 = 5 + 7. For larger even numbers one finds a "tendency" to increase the number of possibilities, but even then there is always a number that has only a few decompositions, such as 98 = 19 + 79 = 31 + 67 = 37 + 61. See the graph below and The On-Line Encyclopedia of Integer Sequences A045917. -
Conjecture of Twin Primes (Still Unsolved Problem in Number Theory) an Expository Essay
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 12 (2017), 229 { 252 CONJECTURE OF TWIN PRIMES (STILL UNSOLVED PROBLEM IN NUMBER THEORY) AN EXPOSITORY ESSAY Hayat Rezgui Abstract. The purpose of this paper is to gather as much results of advances, recent and previous works as possible concerning the oldest outstanding still unsolved problem in Number Theory (and the most elusive open problem in prime numbers) called "Twin primes conjecture" (8th problem of David Hilbert, stated in 1900) which has eluded many gifted mathematicians. This conjecture has been circulating for decades, even with the progress of contemporary technology that puts the whole world within our reach. So, simple to state, yet so hard to prove. Basic Concepts, many and varied topics regarding the Twin prime conjecture will be cover. Petronas towers (Twin towers) Kuala Lumpur, Malaysia 2010 Mathematics Subject Classification: 11A41; 97Fxx; 11Yxx. Keywords: Twin primes; Brun's constant; Zhang's discovery; Polymath project. ****************************************************************************** http://www.utgjiu.ro/math/sma 230 H. Rezgui Contents 1 Introduction 230 2 History and some interesting deep results 231 2.1 Yitang Zhang's discovery (April 17, 2013)............... 236 2.2 "Polymath project"........................... 236 2.2.1 Computational successes (June 4, July 27, 2013)....... 237 2.2.2 Spectacular progress (November 19, 2013)........... 237 3 Some of largest (titanic & gigantic) known twin primes 238 4 Properties 240 5 First twin primes less than 3002 241 6 Rarefaction of twin prime numbers 244 7 Conclusion 246 1 Introduction The prime numbers's study is the foundation and basic part of the oldest branches of mathematics so called "Arithmetic" which supposes the establishment of theorems. -
Elementary Algebra Aei from Wikipedia, the Free Encyclopedia Contents
Elementary algebra aei From Wikipedia, the free encyclopedia Contents 1 Additive identity 1 1.1 Elementary examples ......................................... 1 1.2 Formal definition ........................................... 1 1.3 Further examples ........................................... 1 1.4 Proofs ................................................. 2 1.4.1 The additive identity is unique in a group ........................... 2 1.4.2 The additive identity annihilates ring elements ........................ 2 1.4.3 The additive and multiplicative identities are different in a non-trivial ring .......... 2 1.5 See also ................................................ 2 1.6 References ............................................... 2 1.7 External links ............................................. 3 2 Additive inverse 4 2.1 Common examples .......................................... 4 2.1.1 Relation to subtraction .................................... 4 2.1.2 Other properties ........................................ 4 2.2 Formal definition ........................................... 5 2.3 Other examples ............................................ 5 2.4 Non-examples ............................................. 6 2.5 See also ................................................ 6 2.6 Footnotes ............................................... 6 2.7 References ............................................... 6 3 Algebraic expression 7 3.1 Terminology .............................................. 7 3.2 In roots of polynomials ....................................... -
From China to Paris: 2000 Years Transmission of Mathematical Idea S
FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEA S EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS BOETHIUS TEXTE UND ABHANDLUNGEN ZUR GESCHICHTE DER MATHEMATIK UND DER NATURWISSENSCHAFTEN BEGRIJNDET VON JOSEPH EHRENFRIED HOFMANN FRIEDRICH KLEMM UND BERNHARD STICKER HERAUSGEGEBEN VON MENSO FOLKERTS BAND 46 FRANZ STEINER VERLAG STUTTGART 2002 FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART 2002 Bibliographische Information der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet these Publikation in der Deutschen National- bibliographic; detaillierte bibliographische Daten sind im Internet Uber <http:// dnb.ddb.de> abrufbar. ISBN 3-515-08223-9 ISO 9706 Jede Verwertung des Werkes auBerhalb der Grenzen des Urheberrechtsgesetzes ist unzulhssig and strafbar. Dies gilt insbesondere fair Ubersetzung, Nachdruck. Mikrover- filmung oder vergleichbare Verfahren sowie fi rdie Speicherung in Datenverarbeitungs- anlagen. ® 2002 by Franz Steiner Verlag Wiesbaden GmbH. Sitz Stuttgart. Gedruckt auf s3urefreiem. alterungsbesthndigem Papier. Druck: Druckerei Proff. Eurasburg. Printed in Germany Table of Contents VII Kurt Vogel: A Surveying Problem Travels from China to Paris .................... 1 Jens Hoyrup: Seleucid Innovations in the Babylonian "Algebraic" Tradition and their -
CHINESE, INDIAN, and ARABIC MATHEMATICS 1. Chinese
CHINESE, INDIAN, AND ARABIC MATHEMATICS FRANZ LEMMERMEYER 1. Chinese Mathematics One of the earliest mathematicians of China was Liu Hui (ca. 260 AD). His tools were similar to that of Greek mathematicians, and he even proved theorems; in particular he used similar triangles to solve problems in surveying (e.g. finding the distance between two islands): this is reminiscent of Thales’ use of similar triangles to measure the height of pyramids, or to the method Eratosthenes used for determining the circumference of the earth. He estimated π by approximating the circle with regular n-gons for n = 92 and 184 and knew the principle of exhaustion for circles; he also worked on the volume of the sphere (apparently, the works of Archimedes were not known in China). Zu Chongzhi (429-500) computed π to seven digits. Around 600 AD, Indian works on mathematics were translated into Chinese. Wang Xiaotong (ca. 625) showed how to compute roots of cubic equations nu- merically. Qin Jiushao (1202 – 1261) treated linear systems of congruences (Chinese remainder theorem) and discussed the Euclidean algorithm for computing greatest common divisors. Chinese mathematics had a ‘silver’ period from 300–700, and a ‘golden’ period during the 13th century (the Chinese version of Pascal’s triangle is from this period). Western mathematics was introduced in the 17th century. 2. Indian Mathematics Aryabatha (476–550) introduced and tabulated the sine function, and worked out the solution of linear diophantine equations like ax + by = c, where a, b, c are integers. Although Aryabatha used letter to denote numbers (like the Greeks), he might also have known the decimal system. -
Nptel Course on Mathematics in India: from Vedic Period
NPTEL COURSE ON MATHEMATICS IN INDIA: FROM VEDIC PERIOD TO MODERN TIMES Lecture 40 Mathematics in Modern India 2 M. D. Srinivas Centre for Policy Studies, Chennai 1 Outline I Rediscovering the Tradition (1900-1950) I Rediscovering the Tradition (1950-2010) I Modern Scholarship on Indian Mathematics (1900-2010) I Development of Higher Education in India (1900-1950) I Development of Scientific Research in India (1900-1950) I Development of Modern Mathematics in India (1910-1950) I Development of Modern Mathematics in India (1950-2010) I Development of Higher Education in India (1950-2010) I Halting Growth of Higher Education and Science in India (1980-2010) I Halting Growth of Mathematics in India (1980-2010) 2 Rediscovering the Tradition (1900-1950) Several important texts of Indian mathematics and astronomy were published in the period 1900-1950. Harilal Dhruva published the Rekh¯agan. ita, translation of Euclid from Tusi’s Persian version (Bombay 1901). Vindhyesvari Prasad Dvivedi published some of the ancient siddh¯antas in Jyotis.asiddh¯anta-sa_ngraha (Benares 1912). Babuaji Misra edited the Khan. d. akh¯adyaka of Brahmagupta with Amarja¯ ’s commentary (Calcutta 1925) and Siddh¯anta´sekhara of Sr¯ıpati´ with Makkibhat.t.a’s commentary (Calcutta 1932, 47). Padmakara Dvivedi, edited Gan. itakaumud¯ı of N¯ar¯ayan. a Pan. d. ita in two volumes (1936, 1942). Gopinatha Kaviraja edited the Siddh¯antas¯arvabhauma of Mun¯ı´svara, 2 Vols. (Benares 1933, 3); 3rd Vol. Ed. by Mithalal Ojha (Benres 1978) Kapadia edited the Gan. itatilaka of Sr¯ıdhara´ with commentary (Gaekwad Oriental Series 1935) 3 Rediscovering the Tradition (1900-1950) Several important works were published from the Anand¯a´srama¯ Pune: Karan. -
N N Ln N N Ln N N Ln N N Ln
Practice and Understanding of the Goldbach Problem (A) Tong Xinping (童 信 平) [email protected] Abstract Practice shows: Without enough parameters, 1+2 cannot estimate 1+1 and 1+1×1. At the same time, it is pointed out that among the prime numbers not greater than N, the prime number 6t-1 is more than the prime number 6t+1, (this is the "detail" that Hardy and the others did not notice.) "detail" (= "remainder") affects Represents the negligible oscillation in the accuracy curve of the number r2(N) of Hardy-Litttlewood conjecture (A) in 1921. (In 1989, Hua Luogeng (华罗庚) confirmed that conjecture (A) is the "main term" of even Goldbach's problem.) [Tong Xinping(童信平)Practice and Understanding of theGoldbachProblem(A)AcademArena2021;13(4):71-75]. ISSN1553-992X(print);ISSN2158-771X(online).http://www.sciencepub.net/academia. 7. doi:10.7537/marsaaj130421. 07. Keywords: Practice; Understanding; Goldbach; Problem In 1977, from a logical and philosophical point of view, I thought that "1+2" could not reach "1+1" (short The c(N) in the formula is represented c in the other for even Goldbach conjecture). Starting from interest, I book. am determined to find mathematical evidence that "1+2" Wang Yuan said that to prove "1+1" is to prove cannot reach "1+1" under the guidance of philosophical (1+O(1))→1, which is the "remainder" O(1)→0. The viewpoints. Seeing Xu Chi's article, I dare not act. -0.665 here is a definite value, not →0, and it cannot be However, the passion in my heart cannot be extinguished. -
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Advances in Social Science, Education and Humanities Research, volume 120 World Conference on Management Science and Human Social Development (MSHSD 2017) On the Infiltration of Ideological and Moral Education in Primary Mathematics Teaching Qin Wang1, a and JiangJing Xia 2,b 1 Wuhan University of Technology, Wuhan 430070, China 2 Wuhan University of Technology, Wuhan 430070, China [email protected], [email protected] Keywords: Ideological and moral education, primary school mathematics, infiltration Abstract. Primary school is the initial stage of the formation of a person's outlook on life, values and morality. During this period, the ideological and moral education of students is directly related to the students' ideological and moral standards and personality construction. Primary mathematics teaching not only has a certain role of knowledge inheritance, but also has a certain role in moral education. The infiltration of moral education in mathematics teaching not only improves the ability of mathematical knowledge, but also improves the comprehensive quality of students effectively. According to the students' existing knowledge base, actively looking for the best combination of moral education and mathematics knowledge, well-designed teaching content, grasp the opportunity of teaching, utilizing the characteristics of mathematics to excavate the ideological and moral education contents contained in mathematics, and realizing mathematics teaching and the combination of ideological and moral education is an important issue in primary school mathematics education. 1 Introduction Syllabus for primary school: "when students master basic knowledge, they develop intelligence, improve their ability and receive ideological and moral education.". Primary school mathematics is a very important discipline in elementary education. -
Visiting Mathematicians Jon Barwise, in Setting the Tone for His New Column, Has Incorporated Three Articles Into This Month's Offering
OTICES OF THE AMERICAN MATHEMATICAL SOCIETY The Growth of the American Mathematical Society page 781 Everett Pitcher ~~ Centennial Celebration (August 8-12) page 831 JULY/AUGUST 1988, VOLUME 35, NUMBER 6 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub and annual meetings are joint meetings of the Mathematical Associ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet mathematics and from the headquarters office of the Society. Ab ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions. -
The Nine Chapters on the History of Chinese Mathematics
The Nine Chapters on the History of Chinese Mathematics Rik Brandenburg¤ Keimpe Nevenzeely 15 July 2007 Abstract This article explores Chinese mathematics from the ¯rst archeological evidence of numbers on oracle bones (14th century BC) to the time Chi- nese mathematics became a part universal mathematics (halfway the 19th century AD). First a concise overview of Chinese history and in philosophy is given. The ethical oriented Confucianism was the dominant philosophy and con- sequently little attention was given to the natural world, hindering the development of natural sciences and mathematics. Due to historical and philosophical reasons, Chinese mathematics took quite a di®erent path than its Western counterpart: Chinese mathematics was focused on alge- bra and practical applications instead of geometry and theoretical reason- ing. The Nine Chapters on the Mathematical Art (ca. 1st century AD) is by far the most influential work: it would shape Chinese mathematics for centuries to come. Between the 3rd and the 11th century AD, Bud- dhist and Indian ideas got a ¯rm grip on China, yet curiously Chinese mathematics is barely influenced. In the `Chinese Renaissance' and the subsequent Mongol occupation between the 12th and 14th century Chinese mathematics will reach its zenith. In the 15th and 16th centuries mathematical development waned and important achievements were forgotten. Only after the arrival of Eu- ropean missionary-scientists at the end of the 16th and during the 17th century mathematics made progress again. The Opium Wars of the 19th century mark the end of the classical China and the indigenous Chinese mathematics would be assimilated by universal mathematics. -
An Indian Form of Third Order Taylor Series
HISTORIA MATHEMATICA 1 /1974), 287-289 AN INDIAN FORMOF THIRD ORDERTAYLOR SERIES APPROXIMATIONOF THE SINE BY RADHA CHARAN GUPTA, BIRLA INSTITUTE OF TECHNOLOGY, RANCHI, INDIA SUMMARIES The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation. Approximation formulas for the sine of x + h equivalent to the first two terms of the Taylor series expansion and to the first three terms were known from at least as early as the tenth century A.D. and the fourteenth century A.D. to Indian mathemati- cians [Sengupta 1932, 5; Gupta 1969, 92-941. An approximation formula equivalent to the first four terms, except that the denominator in the fourth term is 4 instead of 6, appears in a literary form in three stanzas quoted by Parameivara (circa 1360-1455) in his supercommentary, called Siddhlnta-dipikz, on Govindasvtiin's gloss (circa 800-850) on the Mahabhaskariya (about early seventh century) [Kuppanna Sastri 1957, 2051. Below we give the text in transliteration from the Siddhlnta- dipik;i and an almost literal translation of the Sanskrit stanzas. Then follows an explanation of the various steps in the rule laid down in the text. (The Indian sine, or Sine, of an angular arc $I is equal to R sin 9, where sin I+ is the modern sine of the angle 4 and R is the radius of the circle of reference).