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Elementary Number Theory

Jie Xiao
International Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a well-done but very limited introduction to elementary number theory, containing just enough material for a one-semester course. The material is severely classical, with everything except RSA encryption being known to Gauss in 1801. A Very Good Feature is the inclusion of many numerical examples. The exercises cover both numerical work and proofs, and all even-numbered problems are answered in the back of the book.

Being so elementary, the book deals primarily with divisibility: primes, unique factorization, solution of linear indeterminate equations, Chinese Remainder Theorem, and the like. But it also covers the irrationality of the square root of 2, quadratic reciprocity, Pythagorean triples, the proof of Fermat’s Last Theorem for exponents 3 and 4, and applications of Gaussian integers.

The books that this one most remind me of are the simple textbooks that came out in the first half of the Twentieth Century, such as Davenport’s The Higher Arithmetic: An Introduction to the Theory of Numbers, Carmichael’s The Theory of Numbers and Diophantine Analysis, and to a lesser extent Ore’s Number Theory and Its History. The present work continues in their tradition, but it is more streamlined and may be more modern in some ways.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • Preface
  • Chapter 1. Basics
    • 1.1. Principles of Induction and Well-ordering
    • 1.2. Equivalence of Principles
    • 1.3. Division Algorithm
    • Exercises
  • Chapter 2. Divisibility
    • 2.1. Divisors
    • 2.2. The Greatest Common Divisor
    • 2.3. The Euclidean Algorithm
    • 2.4. Linear Indeterminate Equation
    • 2.5. The Least Common Multiple
    • Exercises
  • Chapter 3. Primes
    • 3.1. Number of Primes
    • 3.2. The Fundamental Theorem of Arithmetic
    • 3.3. The Bracket Function
    • 3.4. Mersenne, Fermat, and Perfect Numbers
    • Exercises
  • Chapter 4. Congruences
    • 4.1. Simple Properties
    • 4.2. Residue Systems
    • 4.3. Euler’s and Fermat’s Little Theorems
    • 4.4. Orders and Indices
    • Exercises
  • Chapter 5. Congruent Equations
    • 5.1. Linear Congruences and CRT
    • 5.2. Lagrange’s and Wilson’s Theorems
    • 5.3. Quadratic Residues
    • Exercises
  • Chapter 6. Three Additional Topics
    • 6.1. Cryptography by Caesar and RSA
    • 6.2. Diophantine Equations xn + yn = zn, n ≥ 2
    • 6.3. Gaussian Integers
    • Exercises
  • Solutions to Even-numbered Exercises
    • 1. Basics
    • 2. Divisibility
    • 3. Primes
    • 4. Congruences
    • 5. Congruent Equations
    • 6. Three Additional Topics
  • Bibliography
  • Index