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| Introduction | |
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| Factorization and the Primes | |
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| The laws of arithmetic | |
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| Proof by induction | |
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| Prime numbers | |
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| The fundamental theorem of arithmetic | |
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| Consequences of the fundamental theorem | |
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| Euclid's algorithm | |
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| Another proof of the fundamental theorem | |
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| A property of the H.C.F | |
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| Factorizing a number | |
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| The series of primes | |
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| Congruences | |
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| The congruence notation | |
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| Linear congruences | |
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| Fermat's theorem | |
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| Euler's function [phi] (m) | |
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| Wilson's theorem | |
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| Algebraic congruences | |
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| Congruences to a prime modulus | |
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| Congruences in several unknowns | |
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| Congruences covering all numbers | |
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| Quadratic Residues | |
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| Primitive roots | |
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| Indices | |
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| Quadratic residues | |
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| Gauss's lemma | |
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| The law of reciprocity | |
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| The distribution of the quadratic residues | |
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| Continued Fractions | |
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| Introduction | |
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| The general continued fraction | |
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| Euler's rule | |
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| The convergents to a continued fraction | |
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| The equation ax - by = 1 | |
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| Infinite continued fractions | |
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| Diophantine approximation | |
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| Quadratic irrationals | |
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| Purely periodic continued fractions | |
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| Lagrange's theorem | |
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| Pell's equation | |
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| A geometrical interpretation of continued fractions | |
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| Sums of Squares | |
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| Numbers representable by two squares | |
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| Primes of the form 4k + 1 | |
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| Constructions for x and y | |
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| Representation by four squares | |
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| Representation by three squares | |
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| Quadratic Forms | |
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| Introduction | |
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| Equivalent forms | |
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| The discriminant | |
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| The representation of a number by a form | |
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| Three examples | |
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| The reduction of positive definite forms | |
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| The reduced forms | |
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| The number of representations | |
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| The class-number | |
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| Some Diphantine Equations | |
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| Introduction | |
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| The equation x[superscript 2] + y[superscript 2] = z[superscript 2] | |
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| The equation ax[superscript 2] + by[superscript 2] = z[superscript 2] | |
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| Elliptic equations and curves | |
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| Elliptic equations modulo primes | |
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| Fermat's Last Theorem | |
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| The equation x[superscript 3] + y[superscript 3] = z[superscript 3] + w[superscript 3] | |
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| Further developments | |
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| Computers and Number Theory | |
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| Introduction | |
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| Testing for primality | |
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| 'Random' number generators | |
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| Pollard's factoring methods | |
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| Factoring and primality via elliptic curves | |
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| Factoring large numbers | |
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| The Diffie-Hellman cryptographic method | |
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| The RSA cryptographic method | |
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| Primality testing revisited | |
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| Exercises | |
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| Hints | |
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| Answers | |
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| Bibliography | |
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| Index | |