数论概论

Preface

Flowchart ofChapter Dependencies

Introduction

1 What Is Number Theory·

2 Pythagorean Triples

3 Pythagorean Triples and the Unit Circle

4 Sums ofHigher Powe and Fermat’S Last Theorem

5 Divisibility and the Greatest Common Divisor

6 Linear Equatio and the Greatest Common Divisor

7 Factorization and the Fundamental Theorem ofArithmetic

8 Congruences

9 Congruences，Powe ，andFermat’S LittleTheorem

10 Congruences，Powe ，and Euler’S Formula

11 Euler’S Phi Function and the Chinese Remainder Theorem

12 Prime Numbe

13 Counting Primes

14 Me enne Primes

15 Me enne Primes and Perfect Numbe

16 Powe Modulo m and Successive Squaring

17 Computing kth Roots Modulo m

18 Powe ，Roots，and“Unbreakable Codes

l9 Primality Testing and Carmichael Numbe

20 Squares Modulo р

21 Is-l a Square Modulo р·Is 2·

22 Quadratic Reciprocity

23 Proofof Quadratic Reciprocity

24 Which Primes Are Sums ofTwo Squares

25 Which Numbe Are Sums ofTwo Squares

26 As Easy as One，TwO，Three

27 Euler’S Phi Function and Sums of Diviso

28 Powe Modulo р and Primitive Roots

29 Primitive Roots and Indices

30 The Equation X4+Y4=Z

3l Square-Triangular Numbe Revisited

32 Pell’S Equation

33 Diophantine Approximation

34 Diophantine Approximation and Pell’S Equation

35 Number Theory and Imaginary Numbe

36 The Gaussian Intege and Unique Factorization

37 Irrational Numbe and Tra cendental Numbe

38 Binomial Coefficients and Pascal’S Triangle

39 Fibonacci’S Rabbits and Linear Recurrence Sequences

40 Oh，What a Beautiful Function

41 Cubic Curves and Elliptic Curves

42 Elliptic Curves with Few Rational Points

43 Points on Elliptic Curves Modulo р

44 To ion Collectio Modulo р and Bad Primes

45 Defect Bounds and Modularity Patter

46 Elliptic Curves and Fermat’S Last Theorem

Further Reading

Index

47 The Topsy-Turvy World of Continued Fractio [online]

48 Continued Fractio and Pell’s Equation[online]

49 Generating Functio [9nline]

50 Sums of Powe [online]

A Factorization of Small Composite Intege [online]

B A List of Primes [online]

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