Notes to the Reader
1. Divisibility
1.1 Divisors
1.2 Bezout’s identity
1.3 Least common multiples
1.4 Linear Diophantine equations
1.5 Supplementary exercises
2. Prime Numbers
2.1 Prime numbers and prime-power factorisations
2.2 Distribution of primes
2.3 Fermat and Mersenne primes
2.4 Primality-testing and factorisation
2.5 Supplementary exercises
3. Congruences
3.1 Modular arithmetic
3.2 Linear congruences
3.3 Simultaneous linear congruences
3.4 Simultaneous non-linear congruences
3.5 An extension of the Chinese Remainder Theorem
3.6 Supplementary exercises
4. Congruences with a Prime-power Modulus
4.1 The arithmetic of Zp
4.2 Pseudoprimes and Carmiehael numbers
4.3 Solving congruences mod (pe)
4.4 Supplementary exercises
5. EulerTs Function
5.1 Units
5.2 Euler's function
5.3 Applications of Euler's function
5.4 Supplementary exercises
6. The Group of Units
6.1 The group Un
6.2 Primitive roots
6.3 The group Ups, where p is an odd prime
6.4 The group U2
6.5 The existence of primitive roots
6.6 Applications of primitive roots
6.7 The algebraic structure of Un
6.8 The universal exponent
6.9 Supplementary exercises
7. Quadratic Residues
7.1 Quadratic congruences
7.2 The group of quadratic residues
7.3 The Legendre symbol
7.4 Quadratic reciprocity
7.5 Quadratic residues for prime-power moduli
7.6 Quadratic residues for arbitrary moduli
7.7 Supplementary exercises
8. Arithmetic Functions
8.1 Definition and examples
8.2 Perfect numbers
8.3 The MSbius Inversion Formula
8.4 An application of the M6bius Inversion Formula
8.5 Properties of the M6bius function
8.6 The Dirichlet product
8.7 Supplementary exercises
9. The Riemann Zeta Function
9.1 Historical background
9.2 Convergence
9.3 Applications to prime numbers
……
10. Sums of Squares
11. Fermat’s Last Theorem
Appendix A. Induction and Well-ordering
Appendix B. Groups, Rings and Fields
Appendix C. Convergence
Appendix D. Table of Primes p<1000
Solutions to Exercises
Bibliography
Index of symbols
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