An Invitation to Modern Number Theory (Hardback)

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LIKE NEW: This book is in excellent, like-new condition. It was printed on acid-free paper.

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Continued from the back cover: "Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remakes for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

"Stephen Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldback's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

"Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class."

BRIEF CONTENTS

Part 1. Basic Number Theory

1. Mod p Arithmetic, Group Theory and Cryptography
2. Arithmetic Functions
3. Zeta and L-Functions
4. Solutions to Diophantine Equations

Part 2. Continued fractions and Approximations

5. Algebraic and Transcendental Numbers
6. The Proof of Roth's Theorem
7. Introduction to Continued Fractions

Part 3. Probabilistic Methods and Equidistribution

8. Introduction to Probability
9. Applications of Probability: Benford's Law and Hypothesis Testing
10. Distrubiton of Digits of Continued Fractions
11. Introduction to Fourier Analysis
12. {n^kα} and Poissonian Behavior

Part 4. The Circle Method

13. Introduction to the Circle Method
14. Circle Method: Heuristics for Germain Primes

Part 5. Random Matrix Theory and L-Functions

15. From Nuclear Physics to L-Functions
16. Random Matrix Theory: Eigenvalue Densities
17. Random Matrix Theory: Spacings between Adjacent Eigenvalues
18. The Explicit Formula and Density Conjectures

Appendices

1. Analysis Review
2. Linear Algebra Review
3. Hints and Remarks on the Exercises
4. Concluding Remarks

The book ends with Bibliography and Index.