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This book gives an undergraduate-level introduction to Number Theory with the emphasis on fully explained proofs and examples; exercises (with solutions) are integrated into the text. The first few chapters covering divisibility prime numbers and modular arithmetic assume only basic school algebra and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part suitable for third-year students uses ideas from algebra analysis calculus and geometry to study Dirichlet series and sums of squares; in particular the last chapter gives a concise account of Fermat's Last Theorem from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.
Preface
Notes to the reader
Divisibility
Prime Numbers
Congruences
Congruences with prime modulus
Euler's function
The group of units
Quadratic residues
Arithmetic functions
The Riemann zeta function
Sums of squares
Fermat's Last Theorem
Appendix 1: Induction and well-ordering
Appendix 2: Groups rings and fields
Appendix 3: Convergence
Table of primes
Solutions to excercises
References
Index of symbols
Index of names
Index.
This book gives an undergraduate-level introduction to Number Theory with the emphasis on fully explained proofs and examples; exercises (with solutions) are integrated into the text. The first few chapters covering divisibility prime numbers and modular arithmetic assume only basic school algebra and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part suitable for third-year students uses ideas from algebra analysis calculus and geometry to study Dirichlet series and sums of squares; in particular the last chapter gives a concise account of Fermat's Last Theorem from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.
Table of Contents
Preface
Notes to the reader
Divisibility
Prime Numbers
Congruences
Congruences with prime modulus
Euler's function
The group of units
Quadratic residues
Arithmetic functions
The Riemann zeta function
Sums of squares
Fermat's Last Theorem
Appendix 1: Induction and well-ordering
Appendix 2: Groups rings and fields
Appendix 3: Convergence
Table of primes
Solutions to excercises
References
Index of symbols
Index of names
Index.