DOCSLIB.ORG

A Reformulation of the Goldbach Conjecture

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Reformulation of the Goldbach Conjecture 44 MATHEMATICS MAGAZINE A Reformulation of the Goldbach Conjecture LARRY J. GERSTEIN University of California Santa Barbara, CA 93106 The Goldbach conjecture, dating from Goldbach's correspondence with Euler in 1742, is this: Every even integer greater than 2 is the sum of two prime numbers (not necessarily distinct). It has been verified that the even integers through 10' have the stated property, but Goldbach's conjecture remains unproved. The main purpose of this note is to restate Goldbach's conjecture in a somewhat different form. While it will be easy to check that the version to be given here is equivalent to the original, this formulation does not seem to be explicitly stated in the literature. Of course the "literature" on the Goldbach conjecture is vast, and it includes almost every book with "elementary number theory" in its title, as well as a multitude of research papers. For the convenience of the reader, here are a few places to look for references and background information. Dickson [I, pp. 421-4251 lists results obtained through the first decade of this century. Wang [4] is a collection of this century's most important papers on the Goldbach conjecture through the early 1980's. Shanks [3] is a marvelous book on conjectures in number theory, Goldbach's among them, and on mathematical conjectures in general. Ribenboim [2] is another treasure chest in number theory, with a section on Goldbach (pp. 229-235). As a consequence of the reformulation to be given here, we will see a close connection between the Goldbach conjecture and the twin prime conjecture, which is: There are infinitely many primes p such that p + 21 is also prime. (See [2], pp. 199-204, for some history and the current state of affairs regarding the twin prime conjecture.) This connection is certainly not well known, as can be confirmed by the following experiment: Grab some elementary number theory books, locate the inevitable statements of the Goldbach and twin prime problems, and see whether any linkage is established. I'm betting that the only linkages you will find are that both problems deal with prime numbers and both problems are described as famous, old, hard, and open. Suppose the Goldbach conjecture is true. Then for each integer n 2 2 there are primes p and q, with p I q, say, such that 2n = p + q. Hence p = n - k and q = n + k for some integer k satisfying 0 _< k _< n - 2. Therefore n2 - k = pq. Con- versely, suppose that for each integer n 2 2 there exists k satisfying 0 _< k _< n - 2 and primes p and q, with p _< q, such that n2 - k2= pq. Then p = n - k and q = n + k, by the fundamental theorem of arithmetic, and hence 2n = p + q. We have shown therefore that Goldbach's conjecture is equivalent to the following statement: For every integer n 2 2 there exist integers k, p, and q, with (*) 01k_<n-2andwithpandqprime,suchthatn"k~=~q. This restatement of the Goldbach conjecture leads us to consider the binary quadratic form x2 - Y2,and here are some elementary observations. For p and q given odd primes, with p Iq, a straightforward arithmetic argument shows that there are just two nonnegative integer solutions to the equation x2 - Y2= pq; namely, VOL. 66, NO. 1, FEBRUARY 1993 45 On the other hand, there are infinitely many rational solutions. In fact, if P E Q*= Q - {0),then for all A E Q* we have For the Goldbach conjecture we need to work in the opposite direction; that is, we consider the values of x2 - y2 when x = n is fixed and y varies in Z,and we look for appropriate primes p and q. Whether or not (T) is true, it is intriguing to ask how often k = 1 works in (*) and to make the following conjecture: There are infinitely many integers n 2 2 for which there exist primes p (**> and q such that n2- 1 = pq. Statement (**) is clearly equivalent to the twin prime conjecture. Thus, while the Goldbach and twin prime conjectures are not the same, they are evidently facets of the same jewel. REFERENCES 1. Leonard E. Dickson, History of the Theory of Numbers, Vol. 1, Chelsea Publishing Co., New York, 1971. 2. Paulo Ribenboim, The Book of Prime Number Records, 2nd edition, Springer-Verlag Inc., New York, 1989. 3. Daniel Shanks, Solved and Unsolved Problems in Number Theory, 3rd edition, Chelsea Publishing Co., New York, 1985. 4. Wang Yuan (Editor), Goldbach Conjecture, World Scientific Publishing Co., Singapore, 1984. A Lesser-Known Gold bach Conjecture LAURENT HODGES Iowa State University Ames, IA 5001 1 In Example 64 of his recent article [l],R. K. Guy asked about the representations of odd positive integers as sums of a prime and twice a square, mentioning that Ruemmler had found the first exception to be 5777 and wondered if it might be the last. This problem has a long history, beginning with Christian Goldbach, best known for his 1742 conjecture that every even integer can be represented as the sum of two primes. In the last paragraph of a letter to Leonhard Euler dated 18 November 1752, Goldbach expressed his belief that every odd integer could be written in the form p + 2a2,where p is a prime (or 1, then considered a pime) and a 2 0 is an integer [2, p. 5941. Writing in his usual mixture of German and Latin, Goldbach said: Noch habe ein kleines ganz neues theorema beyzufiigen, welches so lange vor wahr halte, donec probetur contrarium: Omnis numerus impar est = duplo quadrati + numero primo, sive 2n - 1 = 2aa + p, ubi a denotet numerum integrum vel 0, p numerum primum, ex. gr. 17 = 2 SO' + 17, 21 = 2.1' + 19, 27 = 2.2' + 19, etc. .
Load more

Recommended publications
  • Leonhard Euler: His Life, the Man, and His Works∗
    SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
    [Show full text]
  • Leonhard Euler Moriam Yarrow
    Leonhard Euler Moriam Yarrow Euler's Life Leonhard Euler was one of the greatest mathematician and phsysicist of all time for his many contributions to mathematics. His works have inspired and are the foundation for modern mathe- matics. Euler was born in Basel, Switzerland on April 15, 1707 AD by Paul Euler and Marguerite Brucker. He is the oldest of five children. Once, Euler was born his family moved from Basel to Riehen, where most of his childhood took place. From a very young age Euler had a niche for math because his father taught him the subject. At the age of thirteen he was sent to live with his grandmother, where he attended the University of Basel to receive his Master of Philosphy in 1723. While he attended the Universirty of Basel, he studied greek in hebrew to satisfy his father. His father wanted to prepare him for a career in the field of theology in order to become a pastor, but his friend Johann Bernouilli convinced Euler's father to allow his son to pursue a career in mathematics. Bernoulli saw the potentional in Euler after giving him lessons. Euler received a position at the Academy at Saint Petersburg as a professor from his friend, Daniel Bernoulli. He rose through the ranks very quickly. Once Daniel Bernoulli decided to leave his position as the director of the mathmatical department, Euler was promoted. While in Russia, Euler was greeted/ introduced to Christian Goldbach, who sparked Euler's interest in number theory. Euler was a man of many talents because in Russia he was learning russian, executed studies on navigation and ship design, cartography, and an examiner for the military cadet corps.
    [Show full text]
  • Goldbach's Conjecture
    Irish Math. Soc. Bulletin Number 86, Winter 2020, 103{106 ISSN 0791-5578 Goldbach's Conjecture: if it's unprovable, it must be true PETER LYNCH Abstract. Goldbach's Conjecture is one of the best-known unsolved problems in mathematics. Over the past 280 years, many brilliant mathematicians have tried and failed to prove it. If a proof is found, it will likely involve some radically new idea or approach. If the conjecture is unprovable using the usual axioms of set theory, then it must be true. This is because, if a counter-example exists, it can be found by a finite search. In 1742, Christian Goldbach wrote a letter to Leonhard Euler proposing that every integer greater than 2 is the sum of three primes. Euler responded that this would follow from the simpler statement that every even integer greater than 2 is the sum of two primes. Goldbach's Conjecture is one of the best-known unsolved problems in mathematics. It is a simple matter to check the conjecture for the first few cases: 4=2+2 6=3+3 8=5+3 10 = 7 + 3 12 = 7 + 5 14 = 11 + 3 16 = 13 + 3 18 = 13 + 5 20 = 17 + 3 ········· But, with an infinite number of cases, this approach can never prove the conjecture. If, for an even number 2n we have 2n = p1 + p2 where p1 and p2 are primes, then obviously p + p n = 1 2 (1) 2 If every even number can be partitioned in this way, then any number, even or odd, must be the arithmetic average, or mean, of two primes.
    [Show full text]
  • 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.
    [Show full text]
  • Linking Together Members of the Mathematical Carlos ­Rocha, University of Lisbon; Jean Taylor, Cour- Community from the US and Abroad
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features Epimorphism Theorem Prime Numbers Interview J.-P. Bourguignon Societies European Physical Society Research Centres ESI Vienna December 2013 Issue 90 ISSN 1027-488X S E European M M Mathematical E S Society Cover photo: Jean-François Dars Mathematics and Computer Science from EDP Sciences www.esaim-cocv.org www.mmnp-journal.org www.rairo-ro.org www.esaim-m2an.org www.esaim-ps.org www.rairo-ita.org Contents Editorial Team European Editor-in-Chief Ulf Persson Matematiska Vetenskaper Lucia Di Vizio Chalmers tekniska högskola Université de Versailles- S-412 96 Göteborg, Sweden St Quentin e-mail: [email protected] Mathematical Laboratoire de Mathématiques 45 avenue des États-Unis Zdzisław Pogoda 78035 Versailles cedex, France Institute of Mathematicsr e-mail: [email protected] Jagiellonian University Society ul. prof. Stanisława Copy Editor Łojasiewicza 30-348 Kraków, Poland Chris Nunn e-mail: [email protected] Newsletter No. 90, December 2013 119 St Michaels Road, Aldershot, GU12 4JW, UK Themistocles M. Rassias Editorial: Meetings of Presidents – S. Huggett ............................ 3 e-mail: [email protected] (Problem Corner) Department of Mathematics A New Cover for the Newsletter – The Editorial Board ................. 5 Editors National Technical University Jean-Pierre Bourguignon: New President of the ERC .................. 8 of Athens, Zografou Campus Mariolina Bartolini Bussi GR-15780 Athens, Greece Peter Scholze to Receive 2013 Sastra Ramanujan Prize – K. Alladi 9 (Math. Education) e-mail: [email protected] DESU – Universitá di Modena e European Level Organisations for Women Mathematicians – Reggio Emilia Volker R. Remmert C. Series ............................................................................... 11 Via Allegri, 9 (History of Mathematics) Forty Years of the Epimorphism Theorem – I-42121 Reggio Emilia, Italy IZWT, Wuppertal University [email protected] D-42119 Wuppertal, Germany P.
    [Show full text]
  • Goldbach's Conjecture
    U.U.D.M. Project Report 2020:37 Goldbach’s Conjecture Johan Härdig Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Augusti 2020 Department of Mathematics Uppsala University Goldbach's Conjecture Johan H¨ardig Contents 1 Introduction 3 1.1 Definition of the Conjectures . 3 2 Prime Numbers and their Distribution 4 2.1 Early Results . 4 2.2 Prime Number Theorem . 6 3 Heuristic and Probabilistic Justification 8 3.1 Method Presented by Gaze & Gaze . 8 3.1.1 Sieve Method by Gaze & Gaze . 9 3.1.2 Example . 10 3.2 Prime Number Theorem for Arithmetic Progressions . 11 3.3 Distribution of Primes Across Prime Residue Classes . 14 3.4 Heuristic Justification by Gaze & Gaze . 15 3.4.1 Conclusion . 16 3.5 Goldbach's Comet . 17 4 The Ternary Goldbach's Conjecture 18 4.1 Historical Overview . 18 4.2 Approach . 19 4.3 Theorems and Methods in the Proof . 20 4.3.1 Hardy-Littlewood Circle Method . 20 4.3.2 Vinogradov's Theorem . 22 4.3.3 The Large Sieve . 24 4.3.4 L-functions . 24 4.3.5 Computational Methods . 26 4.4 The Proof . 29 1 CONTENTS 2 Abstract The following text will provide a historical perspective as well as investigate different approaches to the unsolved mathematical problem Goldbach's conjecture stated by Christian Goldbach in the year 1742. First off, there will be an overview of the early history of prime num- bers, and then a brief description of the Prime Number Theorem. Subsequently, an example of a heuristic and probabilistic method of justifying the binary Goldbach's conjecture, proposed by Gaze and Gaze, will be discussed.
    [Show full text]
  • (Or Seven) Bridges of Kaliningrad: a Personal Eulerian Walk, 2006
    MATCH MATCH Commun. Math. Comput. Chem. 58 (2007) 529-556 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 The Six (or Seven) Bridges of Kaliningrad: a Personal Eulerian Walk, 2006 R. B. Mallion School of Physical Sciences, University of Kent, Canterbury, England, U.K. E-Mail Address: [email protected] (Received June 1, 2007) Abstract The eighteenth-century problem of the Bridges of Königsberg was solved in a memoir dated 1736 and written by the Swiss mathematician Leonhard Euler (1707í1783) soon after he had been appointed to the senior Chair of Mathematics at the St. Petersburg Academy of Sciences. Euler demonstrated that what is now called an Eulerian Walk (that is, a route that traverses all of the bridges once, and once only) was not possible in contemporary Königsberg. Soon after the Conferences of Yalta and Potsdam had assigned the city and its environs to the Soviet Union after World War II, Königsberg came to be known as the city of Kaliningrad (Ʉɚɥɢɧɢɧɝɪɚɞ), capital of the Kaliningrad Oblast, which, since the early 1990s, has found itself as an exclave of the present-day Russian Federation, isolated from mainland Russia by the newly independent republic of Lithuania (and, beyond that, Latvia and Belarus). Furthermore, the Kaliningrad Oblast’s only other adjoining neighbour is Poland which, like Lithuania, has been a Member of the European Union since 1 May 2004. This state of affairs thus determines that the Kaliningrad Oblast is, these days, doubly anomalous, in that it is not only an exclave of the Russian Federation but (simultaneously) it is also a foreign enclave within the European Union.
    [Show full text]
  • AN EXPLORATION on GOLDBACH's CONJECTURE E. Markakis1, C
    International Journal of Pure and Applied Mathematics Volume 84 No. 1 2013, 29-63 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v84i1.3 ijpam.eu AN EXPLORATION ON GOLDBACH’S CONJECTURE E. Markakis1, C. Provatidis2 §, N. Markakis3 1Vassilissis Olgas 129B 54643, Thessaloniki, GREECE 2National Technical University of Athens 9, Heroes of Polytechnion Ave., Zografou Campus 157 80, Athens, GREECE 3Cram School “Methodiko” Vouliagmenis and Kyprou 2, 16452 Argyroupolis, GREECE Abstract: This paper divides the set of natural numbers in six equivalence classes and determines two of them as candidate to include all prime numbers. Concerning the even numbers themselves, these were divided into three subsets using a basic cell (6n 2, 6n and 6n + 2). Based on the aforementioned tools, − this paper proposes a deterministic process of finding all pairs (p, q) of odd numbers (composites and primes) of natural numbers 3 whose sum (p + q) is ≥ equal to a given even natural number 2n 6. Based on this procedure and also ≥ relying on the distribution of primes in the set N of natural numbers, a closed analytical formula is proposed for the estimation of the number of primes that satisfy Goldbach’s conjecture for positive integers 6. ≥ AMS Subject Classification: 11-XX Key Words: Goldbach’s conjecture, prime numbers, statistics c 2013 Academic Publications, Ltd. Received: September 27, 2012 url: www.acadpubl.eu §Correspondence author 30 E. Markakis, C. Provatidis, N. Markakis 1. Introduction As is known, on June 7, 1742, Christian Goldbach in a letter to Leonhard Euler (see [1]) argued that, every even natural number greater than 4 can be written as a sum of two primes, namely: 2n = p + q, where n > 2, and p, q are prime numbers.
    [Show full text]
  • A Lesser-Known Gold Bach Conjecture
    VOL. 66, NO. 1, FEBRUARY 1993 45 On the other hand, there are infinitely many rational solutions. In fact, if P E Q*= Q - {0),then for all A E Q* we have For the Goldbach conjecture we need to work in the opposite direction; that is, we consider the values of x2 - y2 when x = n is fixed and y varies in Z,and we look for appropriate primes p and q. Whether or not (*) is true, it is intriguing to ask how often k = 1 works in (*) and to make the following conjecture: There are infinitely many integers n 2 2 for which there exist primes p (**> and q such that n2- 1 = pq. Statement (**) is clearly equivalent to the twin prime conjecture. Thus, while the Goldbach and twin prime conjectures are not the same, they are evidently facets of the same jewel. REFERENCES 1. Leonard E. Dickson, History of the Theory of Numbers, Vol. 1, Chelsea Publishing Co., New York, 1971. 2. Paulo Ribenboim, The Book of Prime Number Records, 2nd edition, Springer-Verlag Inc., New York, 1989. 3. Daniel Shanks, Solved and Unsolved Problems in Number Theory, 3rd edition, Chelsea Publishing Co., New York, 1985. 4. Wang Yuan (Editor), Goldbach Conjecture, World Scientific Publishing Co., Singapore, 1984. A Lesser-Known Gold bach Conjecture LAURENT HODGES Iowa State University Ames, IA 5001 1 In Example 64 of his recent article [l],R. K. Guy asked about the representations of odd positive integers as sums of a prime and twice a square, mentioning that Ruemmler had found the first exception to be 5777 and wondered if it might be the last.
    [Show full text]
  • Euler Transcript
    Euler Transcript Date: Wednesday, 6 March 2002 - 12:00AM Euler Professor Robin Wilson "Read Euler, read Euler, he is the master of us all...." So said Pierre-Simon Laplace, the great French applied mathematician. Leonhard Euler, the most prolific mathematician of all time, wrote more than 500 books and papers during his lifetime about 800 pages per year with another 400 publications appearing posthumously; his collected works already fill 73 large volumes tens of thousands of pages with more volumes still to appear. Euler worked in an astonishing variety of areas, ranging from the very pure the theory of numbers, the geometry of a circle and musical harmony via such areas as infinite series, logarithms, the calculus and mechanics, to the practical optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides. Euler originated so many ideas that his successors have been kept busy trying to follow them up ever since. Many concepts are named after him, some of which Ill be talking about today: Eulers constant, Eulers polyhedron formula, the Euler line of a triangle, Eulers equations of motion, Eulerian graphs, Eulers pentagonal formula for partitions, and many others. So with all these achievements, why is it that his musical counterpart, the prolific composer Joseph Haydn, creator of symphonies, concertos, string quartets, oratorios and operas, is known to all, while he is almost completely unknown to everyone but mathematicians? Today, Id like to play a small part in restoring the balance by telling you about Euler, his life, and his diverse contributions to mathematics and the sciences.
    [Show full text]
  • ABSTRACT on the Goldbach Conjecture Westin King Director: Dr
    ABSTRACT On the Goldbach Conjecture Westin King Director: Dr. David Arnold, Ph.D. Goldbach's conjecture, proposed in 1742, is one of the oldest open questions in mathematics. Much work has been done on the problem, and despite significant progress, a solution remains elusive. The goal of this paper is to give an introduction to Goldbach's conjecture, discuss the history of the problem, summarize important papers on the subject, examine methodologies used to attack the problem, and explain related problems and consequences of the conjecture. APPROVED BY DIRECTOR OF HONORS THESIS: __________________________________________ Dr. David Arnold, Department of Mathematics APPROVED BY THE HONORS PROGRAM: __________________________________________ Dr. Andrew Wisely, Director DATE: _______________________ ON THE GOLDBACH CONJECTURE A Thesis Submitted to the Faculty of Baylor University In Partial Fulfillment of the Requirements for the Honors Program By Westin King Waco, Texas May 2013 TABLE OF CONTENTS Acknowledgments iii CHAPTER ONE: Introduction 1 Biography of Christian Goldbach 3 History 3 CHAPTER TWO: Methods of Proof 6 Hardy-Littlewood Circle Method 6 Selberg Sieve 7 The Riemann Hypothesis 10 Computing 12 CHAPTER THREE: Summaries of Major Papers 14 On the Vinogradov bound in the three primes Goldbach conjecture 14 On Šnirel’man’s Constant 14 All odd integers greater than one are the sum of at most five primes 15 On Vinogradov’s constant in Goldbach’s ternary problem 16 On Šnirelman’s constant under the Riemann hypothesis 18 CHAPTER FOUR: Related Conjectures and Consequences 19 Twin Primes Conjecture 19 Goldbach’s Number 20 Untouchable Numbers 22 CHAPTER FIVE: Outlook 24 References 25 ii ACKNOWLEDGMENTS The author would like to thank Dr.
    [Show full text]
  • The Impact of the Relationship Between Peter the Istand Leibniz On
    The impact of the relationship between Peter I and Leibniz on the development of science in Russia M. Rosa Massa-Esteve Centre de Recerca per a la Història de la Tècnica. Departament de Matemàtiques. Universitat Politècnica de Catalunya. Abstract: Peter the Great (1672-1725), Tsar of Russia, travelled to Western Europe specifically to German Lands, to familiarize himself with new developments and to contact with men of science, like Gottfried Wilhelm Leibniz (1646-1716). The aim of this article is to analyze Peter the Ist relationship with Leibniz, in order to understand better how and in what way the influence of Leibniz’s ideas was reflected in the formation and later in the development of Saint Petersburg Academy of Sciences and Arts. The analysis of their correspondence, of meetings and of the projects (memoranda) that Leibniz presented to the Tsar for the modernization of Russia enable us to know better Leibniz’s influences on the creation of the Academy of Sciences in Saint Petersburg as well as on the ideas for Tsar’s Academy, such as the universality of knowledge and the meaning of scientific knowledge as theory for practice, Theoria cum praxi, also lemma for Leibniz’s Berlin Academy. Keywords: Peter the Ist ; Leibniz; Saint Petersburg Academy; eighteenth century; Theoria cum praxi; Russia. Introduction1 Peter the Great (1672-1725), Tsar of Russia, founded the Saint Petersburg Academy of Sciences and Arts [18 and 8],2which became the major scientific centre of the country and has played a key role in the transfer and circulation of knowledge between European science and Russia.
    [Show full text]