| Preface |
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ix | |
| Contents |
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xi | |
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xiii | |
| Abbreviations, basic references and notations |
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xvii | |
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1 | (3) |
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4 | (2) |
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6 | (1) |
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6 | (1) |
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7 | (3) |
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10 | (2) |
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Fibonacci and the Liber Quadratorum |
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12 | (2) |
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Early work on Pell's equation |
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14 | (3) |
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Pell's equation: Archimedes and the Indians |
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17 | (7) |
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Diophantus and diophantine equations |
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24 | (5) |
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Diophantus and sums of squares |
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29 | (2) |
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Diophantus's resurgence: Viete and Bachet |
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31 | (6) |
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Fermat and his Correspondents |
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37 | (9) |
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46 | (3) |
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49 | (2) |
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Perfect numbers and Fermat's theorem |
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51 | (8) |
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59 | (2) |
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First attempts on quadratic residues |
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61 | (2) |
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The prime divisors of sums of two squares |
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63 | (3) |
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66 | (3) |
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Numbers of representations by sums of two squares |
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69 | (6) |
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Infinite descent and the equation x4 - y4 = z2 |
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75 | (4) |
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The problems of Fermat's maturity |
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79 | (4) |
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``Elementary'' quadratic forms |
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83 | (9) |
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92 | (8) |
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Indeterminate equations of degree 2 |
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100 | (3) |
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The ascent for equations of genus 1 |
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103 | (9) |
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112 | (6) |
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118 | (41) |
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Appendix I: Euclidean quadratic fields |
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125 | (5) |
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Appendix II: Curves of genus 1 in projective spaces |
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130 | (5) |
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Appendix III: Fermat's ``double equations'' as space quartics |
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135 | (5) |
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Appendix IV: The descent and Mordell's theorem |
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140 | (10) |
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Appendix V: The equation y2 = x3 - 2x |
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150 | (9) |
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Scientific life in the sixteenth, seventeenth and eighteenth century |
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159 | (3) |
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162 | (7) |
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169 | (3) |
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Euler's discovery of number-theory |
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172 | (4) |
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176 | (13) |
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The multiplicative group modulo N |
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189 | (12) |
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``Real'' vs. ``imaginary'' |
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201 | (3) |
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The missing quadratic reciprocity law |
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204 | (6) |
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210 | (9) |
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The search for large primes |
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219 | (7) |
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226 | (3) |
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Square roots and continued fractions |
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229 | (4) |
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Diophantine equations of degree 2 |
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233 | (6) |
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More diophantine equations |
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239 | (3) |
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Elliptic integrals and the addition theorem |
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242 | (10) |
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Elliptic curves as diophantine equations |
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252 | (4) |
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The summation formula and Σn-ν |
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256 | (5) |
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Euler and the zeta-function |
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261 | (6) |
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The trigonometric functions |
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267 | (5) |
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The functional equation for the zeta-function |
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272 | (4) |
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Partitio numerorum and modular functions |
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276 | (7) |
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283 | (26) |
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Appendix I: The quadratic reciprocity law |
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287 | (5) |
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Appendix II: An elementary proof for sums of squares |
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292 | (4) |
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Appendix III: The addition theorem for elliptic curves |
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296 | (13) |
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An Age of Transition: Lagrange and Legendre |
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309 | (5) |
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Lagrange and number theory |
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314 | (2) |
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316 | (2) |
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Lagrange's theory of binary quadratic forms |
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318 | (4) |
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322 | (4) |
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Legendre's arithmetical work |
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326 | (35) |
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Appendix I: Hasse's principle for ternary quadratic forms |
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339 | (7) |
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Appendix II: A proof of Legendre's on positive binary quadratic forms |
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346 | (4) |
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Appendix III: A proof of Lagrange's on indefinite binary quadratic forms |
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350 | (11) |
| Additional bibliography and references |
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361 | (4) |
| Index nominum |
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365 | (7) |
| Index rerum |
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372 | |
