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Elementary Number Theory

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Elementary Number Theory

The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by non-majors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. The exercises are carefully chosen to broaden the understanding of the concepts.

Combinatorics and Discrete MathematicsMathematics
Author
Wissam Raji
Publisher
American University of Beirut
Source
LibreTexts
License
CC BY
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 49 sections
  1. 01: Introduction
  2. 1.01: Algebraic Operations With Integers
  3. 1.02: The Well Ordering Principle and Mathematical Induction
  4. 1.03: Divisibility and the Division Algorithm
  5. 1.04: Representations of Integers in Different Bases
  6. 1.05: The Greatest Common Divisor
  7. 1.06: The Euclidean Algorithm
  8. 1.07: Lame's Theorem
  9. 02: Prime Numbers
  10. 2.01: The Sieve of Eratosthenes
  11. 2.02: The Infinitude of Primes
  12. 2.03: The Fundamental Theorem of Arithmetic
  13. 2.04: Least Common Multiple
  14. 2.05: Linear Diophantine Equations
  15. 2.06: The function x. the symbols O o and
  16. 2.07: Theorems and Conjectures involving prime numbers
  17. 03: Congruences
  18. 3.01: Introduction to Congruences
  19. 3.02: Residue Systems and Eulers -Function
  20. 3.03: Linear Congruences
  21. 3.04: The Chinese Remainder Theorem
  22. 3.05: Theorems of Fermat Euler and Wilson
  23. 04: Multiplicative Number Theoretic Functions
  24. 4.01: Definitions and Properties
  25. 4.02: Multiplicative Number Theoretic Functions
  26. 4.03: The Mobius Function and the Mobius Inversion Formula
  27. 4.04: Perfect Mersenne and Fermat Numbers
  28. 05: Primitive Roots and Quadratic Residues
  29. 5.01: The order of Integers and Primitive Roots
  30. 5.02: Primitive Roots for Primes
  31. 5.03: The Existence of Primitive Roots
  32. 5.04: Introduction to Quadratic Residues and Nonresidues
  33. 5.05: Legendre Symbol
  34. 5.06: The Law of Quadratic Reciprocity
  35. 5.07: Jacobi Symbol
  36. 06: Introduction to Continued Fractions
  37. 6.01: Basic Notations
  38. 6.02: Main Technical Tool
  39. 6.03: Very Good Approximation
  40. 6.04: An Application
  41. 6.05: A Formula of Gauss a Theorem of Kuzmin and Levi and a Problem of Arnold
  42. 07: Introduction to Analytic Number Theory
  43. 7.01: Introduct to Analytic Number Theory
  44. 7.02: Chebyshev's Functions
  45. 7.03: Getting Closer to the Proof of the Prime Number Theorem
  46. 08: Other Topics in Number Theory
  47. 8.01: Cryptography
  48. 8.02: Elliptic Curves
  49. 8.03: The Riemann Zeta Function