TY  - BOOK
AU  - Jones,Gareth A.
AU  - Jones,Josephine M.
TI  - Elementary Number Theory
T2  - Springer Undergraduate Mathematics Series
SN  - 9781447106135
AV  - QA241-247.5 
U1  - 512/.7 
PY  - 1998///
CY  - London
PB  - Springer London, Limited
KW  - Number theory
KW  - Electronic books
N1  - Cover -- Title -- Copyright -- Preface -- Contents -- 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 Carmichael numbers -- 4.3 Solving congruences mod (pe) -- 4.4 Supplementary exercises -- 5 Euler's 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 Upe, where p is an odd prime -- 6.4 The group U2e -- 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 Möbius Inversion Formula -- 8.4 An application of the Möbius Inversion Formula -- 8.5 Properties of the Möbius 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 -- 9.4 Random integers -- 9.5 Evaluating '(2) -- 9.6 Evaluating (2k) -- 9.7 Dirichlet series -- 9.8 Euler products -- 9.9 Complex variables -- 9.10 Supplementary exercises -- 10 Sums of Squares -- 10.1 Sums of two squares -- 10.2 The Gaussian integers -- 10.3 Sums of three squares -- 10.4 Sums of four squares -- 10.5 Digression on quaternions -- 10.6 Minkowski's Theorem -- 10.7 Supplementary exercises -- 11. Fermat's Last Theorem -- 11.1 The problem -- 11.2 Pythagoras's Theorem -- 11.3 Pythagorean triples -- 11.4 Isosceles triangles and irrationality -- 11.5 The classification of Pythagorean triples -- 11.6 Fermat -- 11.7 The case n = 4 -- 11.8 Odd prime exponents -- 11.9 Lamé and Kummer -- 11.10 Modern developments -- 11.11 Further reading -- Appendix A. Induction and Well-ordering -- Appendix B. Groups, Rings and Fields -- Appendix C. Convergence -- Appendix D. Table of Primes p &lt -- 1000 -- Solutions to Exercises -- Bibliography -- Index of symbols -- Index of names -- Index
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