Elementary Number Theory

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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