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Number Theory: An approach through history From Hammurapi to Legendre 2001 ed. [Minkštas viršelis]

4.45/5 (21 ratings by Goodreads)
  • Formatas: Paperback / softback, 377 pages, aukštis x plotis: 235x155 mm, weight: 1250 g, 19 Illustrations, black and white; XXI, 377 p. 19 illus., 1 Paperback / softback
  • Serija: Modern Birkhauser Classics
  • Išleidimo metai: 22-Dec-2006
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 0817645659
  • ISBN-13: 9780817645656
Kitos knygos pagal šią temą:
  • Formatas: Paperback / softback, 377 pages, aukštis x plotis: 235x155 mm, weight: 1250 g, 19 Illustrations, black and white; XXI, 377 p. 19 illus., 1 Paperback / softback
  • Serija: Modern Birkhauser Classics
  • Išleidimo metai: 22-Dec-2006
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 0817645659
  • ISBN-13: 9780817645656
Kitos knygos pagal šią temą:
Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.



André Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work from an Old Babylonian tablet, datable to the time of Hammurapi to Legendres Essai sur la Théorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

Recenzijos

"The book makes a fascinating reading, permitting to perceive the birth of new ideas, and to understand why they should have been born... There are four chapters: Protohistory, Fermat and his correspondents, Euler and An age of transition: Lagrange and Legendre, and also several appendices, which introduce a modern point of view and provide proofs for many mentioned results. The book is strongly recommended to anybody interested in the history of mathematics and should be on the shelf of every number-theorist."





--Zentralblatt Math



"As the author says, this is a historical treatment of that oldest and purest field of mathematics, the theory of numbers; his presentation is meticulous and scholarly... The volume under review...is a discursive, expository, leisurely peek over the shoulders of several great authors in number theory, a subject "conspicuous for the quality rather than for the number of its devotees; at the same time it is perhaps unique in the enthusiasm it has inspired", as Professor Weil says in his preface."



--Mathematical Reviews



A very unusual book combining thorough philological exactness, keen observation, apt comments of the essential points, picturesque fantasy, enthusiastic love of the subject, and brilliant literary style: a romantic novel of documents. It is both number theory and its history in an inseparable oneness, helping us understand the very roots and the first big stage of progress of this discipline. The author, one of the most prominent number theoristschose to give us a broad perspective of the birth of modern number theory.



--Periodica Mathematica Hungaria

Daugiau informacijos

Springer Book Archives
Preface ix
Contents xi
Table of illustrations
xiii
Abbreviations, basic references and notations xvii
Protohistory
Introduction
1(3)
Primes and factoring
4(2)
Perfect numbers
6(1)
First degree problems
6(1)
Pythagorean triangles
7(3)
Sums of two squares
10(2)
Fibonacci and the Liber Quadratorum
12(2)
Early work on Pell's equation
14(3)
Pell's equation: Archimedes and the Indians
17(7)
Diophantus and diophantine equations
24(5)
Diophantus and sums of squares
29(2)
Diophantus's resurgence: Viete and Bachet
31(6)
Fermat and His Correspondents
Biographical
37(9)
Binomial coefficients
46(3)
Proofs vs. ``induction''
49(2)
Perfect numbers and Fermat's theorem
51(8)
Early gropings
59(2)
First attempts on quadratic residues
61(2)
The prime divisors of sums of two squares
63(3)
Sums of two squares
66(3)
Numbers of representations by sums of two squares
69(6)
Infinite descent and the equation x4 - y4 = z2
75(4)
The problems of Fermat's maturity
79(4)
``Elementary'' quadratic forms
83(9)
Pell's equation
92(8)
Indeterminate equations of degree 2
100(3)
The ascent for equations of genus 1
103(9)
More on the descent
112(6)
Conclusion
118(7)
Appendix I: Euclidean quadratic fields
125(5)
Appendix II: Curves of genus 1 in projective spaces
130(5)
Appendix III: Fermat's ``double equations'' as space quartics
135(5)
Appendix IV: The descent and Mordell's theorem
140(10)
Appendix V: The equation y2 = x3 -- 2x
150(9)
Euler
Scientific life in the sixteenth, seventeenth and eighteenth century
159(3)
Euler's life
162(7)
Euler and Goldbach
169(3)
Euler's discovery of number-theory
172(4)
Dramatis personae
176(13)
The multiplicative group modulo N
189(12)
``Real'' vs. ``imaginary''
201(3)
The missing quadratic reciprocity law
204(6)
Binary quadratic forms
210(9)
The search for large primes
219(7)
Sums of four squares
226(3)
Square roots and continued fractions
229(4)
Diophantine equations of degree 2
233(6)
More diophantine equations
239(3)
Elliptic integrals and the addition theorem
242(10)
Elliptic curves as diophantine equations
252(4)
The summation formula and Σn-v
256(5)
Euler and the zeta-function
261(6)
The trigonometric functions
267(5)
The functional equation for the zeta-function
272(4)
Partitio numerorum and modular functions
276(7)
Conclusion
283(4)
Appendix I: The quadratic reciprocity law
287(5)
Appendix II: An elementary proof for sums of squares
292(4)
Appendix III: The addition theorem for elliptic curves
296(13)
An Age of Transition: Lagrange and Legendre
Lagrange's life
309(5)
Lagrange and number theory
314(2)
Indeterminate equations
316(2)
Lagrange's theory of binary quadratic forms
318(4)
Legendre's life
322(4)
Legendre's arithmetical work
326(13)
Appendix I: Hasse's principle for ternary quadratic forms
339(7)
Appendix II: A proof of Legendre's on positive binary quadratic forms
346(4)
Appendix III: A proof of Lagrange's on indefinite binary quadratic forms
350(11)
Additional bibliography and references 361(4)
Index nominum 365(7)
Index rerum 372