Elementary Number Theory

Title
Preface
Contents
Chapter I: Elementary properties of integers
1. Origin of the Theory of Numbers
2. Operations of Addition and Multiplication
3. Summation of Certain Series
4. Polygonal Numbers
5. Subtraction and Division
6. Scales of Notation
7. The Binary System
8. The Lure of Number Theory
Exercises and Problems
Chapter II: Divisibility and related topics
1. Theorems Concerning Divisibility
2. Common Divisors
3. Euclid’s Algorithm
4. Greatest Common Divisor of More Than Two Numbers
5. Theorems Concerning Common Divisors
6. The Fundamental Theorem of Arithmetic
7. Common Multiples
8. Solution of x^2 + y^2 = z^2 in Integers
Exercises and Problems
Chapter III: Euclid's algorithm and Diophantine equations of the first degree
1. Lamé’s Theorem
2. The Least-remainder Algorithm
3. Lemma
4. Kronecker’s Theorem
Exercises and Problems
5. Indeterminate Equations of the First Degree
6. Continuation
7. Nonnegative Solutions of Linear Indeterminate Equations
8. Equations in Several Unknowns and Systems of Equations
Exercises and Problems
Chapter IV: On prime numbers
1. Prime and Composite Numbers
Exercises
2. A Test of Primality
3. The Sieve of Eratosthenes
Exercises
4. Unique Factorization Theorem
5. Criterion of Divisibility
6. Divisors of Numbers
7. The Number and Sum of Divisors
8. Numerical Functions Depending on Divisors
9. Euler’s Recurrence Formula
10. Perfect Numbers
Exercises and Problems
11. Number of Primes Infinite
12. Bonse’s Inequality
13. A Property of 30
14. Remarks on the Distribution of Primes
15. Primes in Arithmetic Progressions
16. Some Unsolved Problems Concerning Primes
Exercises and Problems
17. Integral Part of a Real Number
Exercises and Problems
18. The Highest Power of a Prime Contained in a Factorial
19. Some Applications
Exercises and Problems
Chapter V: A general combinatorial theorem and its applications
1. Combinatorial Theorem
2. Euler’s Function phi(n)
3. Moebius’s Function mu(n)
Exercises and Problems
4. Fundamental Property of mu(n)
Exercises and Problems
5. A Property of phi(n)
6. Inversion Formula
Exercises and Problems
7. Another Application of the Combinatorial Formula
8. Meissel’s Formula
Exercises and Problems
Chapter VI: On the congruence of numbers
1. Definition and Simple Properties of Congruences
Exercises
2. Elementary Properties of Congruences Continued
3. Distribution of Numbers in Classes Modulo m
4. Various Useful Complete Systems of Residues
Exercises and Problems
5. Generation of Complete System Modulo ab When (a, b) = 1
6. Generation of Complete System of Residues Mod a^n
7. An Application
8. Reduced System of Residues
Exercises and Problems
9. Theorems of Fermat and Euler
10. Another Proof
Exercises and Problems
11. Residues of S_n(p) mod p
12. Wilson’s Theorem
Exercises and Problems
Appendix: On Magic Squares
1. Definition of Magic Squares
2. Auxiliary Squares
3. Magic Squares for Odd n
4. Magic Squares for n Divisible by 4
5. Magic Squares for n Divisible by 2 Only
Chapter VII: Congruences with one unknown. Lagrange's theorem and its applications
1. Congruences in General
Exercises
2. Congruences of the First Degree
3. Methods for Solving Congruences of the First Degree
4. Continuation
Exercises and Problems
5. An Important System of Congruences
6. Case of Moduli Relatively Prime in Pairs
Exercises
7. Congruences of Higher Degree: Composite Moduli
8. Congruences of Higher Degree: Moduli Powers of Primes
Exercises and Problems
9. Congruences with a Prime Modulus: Lagrange’s Theorem
10. Some Applications of Lagrange’s Theorem
11. Condition for a Congruence to Have Number of Roots Equal to Its Degree
12. An Application
Exercises and Problems
Appendix: Calendar Problems
1. Relation between Dates and Days of the Week
2. Remarks on the Church Calendar
3. The Date of Easter
Chapter VIII: Residues of powers
1. Exponent of a Modulo m
2. Practical Rule for the Formation of Periods
3. Properties of Exponents Modulo m
Exercises and Problems
4. Primitive Roots for Prime Moduli
5. Method for Finding Primitive Roots
Exercises and Problems
6. Indices
7. Application of Indices to the Solution of Congruences
8. Tables of Indices and Primitive Roots
Exercises and Problems
Appendix: On Card Shuffling
1.
2.
Chapter IX: Arithmetical properties of Bernoullian numbers
1. Origin of Bernoullian Numbers
2. Definition of Bernoullian Numbers by a Symbolic Formula
3. The General Expression for the Sum S_n(N)
4. Proof That Bernoullian Numbers Are Positive
5. Staudt’s Theorem
6. An Auxiliary Congruence
7. Another Auxiliary Congruence
8. Voronoi’s Theorem and Its Applications
9. Fractions mod m. Kummer’s Congruences
Exercises and Problems
Chapter X: Quadratic residues
1. Definition of Quadratic Residues
2. Prime Moduli
3. Quadratic Residuality of a Product
4. Euler’s Criterion
5. Legendre’s Symbol
6. Fundamental Problems
7. Quadratic Character of -1 and 2
Exercises and Problems
8. Quadratic Reciprocity Law
9. The Lemma of Gauss
10. Proof of the Reciprocity Law
11. Applications
12. Jacobi’s Symbol
13. The Evaluation of Jacobi’s Symbol
14. Solution of the Equation (P/Q) = -+1 for Q
Exercises and Problems
15. Quadratic Residues of Composite Moduli
16. Moduli 2^m
17. General Conclusion
18. Solution of Quadratic Congruences for Prime Moduli
19. The Exclusion Method
Exercises and Problems
Chapter XI: Some problems connected with quadratic forms
1. Object of This Chapter
2. Fundamental Lemma
3. The Equation x^2 - ay^2 = m
4. Application of the Fundamental Lemma
Exercises and Problems
5. Fermat’s Equation
6. The Equation x^2 - y^2 = m with Positive a
7. A Test of Primality
8. The Exclusion Method
Exercises and Problems
9. Another Application of the Fundamental Lemma
10. Kummer’s Proof of the Reciprocity Law
11. The Four Squares Theorem
Chapter XII: Some Diophantine problems
1. Object of This Chapter
2. Equations x^2 + ay^2 = z^n
3. Particular Cases
4. Some Equations of the Type x^2 + c = y^3
5. Some Insoluble Diophantine Problems
6. Another Fermat Problem
7. Fermat’s Last Theorem
Exercises and Problems
8. One More Fermat Problem
9. An Ancient Problem
Exercises and Problems
Chapter XIII: Liouville's methods
1. Object of This Chapter
2. Arbitrary Functions. Conditions of Parity
3. The First Fundamental Identity
4. The Second Fundamental Identity
5. Euler’s Recurrence Formula
6. Specialization of the Fundamental Identities
7. An Application
8. Jacobi’s Theorem
9. Additional Identities
10. Representations by the Sums of Three Squares
Numerical tables
Index