Here are 100 chapter titles for a book on Number Theory, progressing from beginner to advanced:
I. Foundations and Elementary Number Theory (20 Chapters)
- Introduction to Number Theory: What and Why?
- The Integers: Basic Properties
- Divisibility and the Division Algorithm
- Greatest Common Divisor (GCD) and Euclidean Algorithm
- Least Common Multiple (LCM)
- Prime Numbers and Prime Factorization
- The Fundamental Theorem of Arithmetic
- Congruences: Basic Properties
- Modular Arithmetic
- Linear Congruences and the Chinese Remainder Theorem
- Fermat's Little Theorem and Euler's Theorem
- Wilson's Theorem
- Applications of Congruences: Cryptography (RSA)
- Arithmetic Functions: φ(n), σ(n), d(n)
- Perfect Numbers
- Mersenne Primes and Fermat Numbers
- The Distribution of Primes: Introduction
- The Prime Number Theorem (Intuitive Idea)
- Diophantine Equations: Introduction
- Basic Proof Techniques in Number Theory
II. Classical Number Theory (30 Chapters)
- Quadratic Congruences
- Quadratic Reciprocity
- The Legendre Symbol and Jacobi Symbol
- Primitive Roots
- Indices and Discrete Logarithms
- Continued Fractions: Basic Properties
- Finite and Infinite Continued Fractions
- Applications of Continued Fractions: Approximating Irrationals
- Diophantine Approximation
- Liouville Numbers and Transcendental Numbers
- Algebraic Numbers and Algebraic Integers
- Number Fields: Introduction
- Rings of Integers
- Unique Factorization in Number Fields (Failure and Examples)
- Ideal Theory: Introduction
- Ideals and Factorization in Number Fields
- The Class Number
- Dirichlet's Theorem on Primes in Arithmetic Progressions (Introduction)
- The Riemann Zeta Function: Introduction
- Properties of the Riemann Zeta Function
- The Prime Number Theorem (Proof Sketch)
- The Riemann Hypothesis (Statement and Significance)
- Elliptic Curves: Introduction
- Elliptic Curves and Cryptography
- The Birch and Swinnerton-Dyer Conjecture (Statement)
- Modular Forms: Introduction
- Modular Forms and Elliptic Curves
- The Taniyama-Shimura Theorem (Statement)
- Fermat's Last Theorem (History and Proof Idea)
- The ABC Conjecture (Statement and Implications)
III. Modern Number Theory (30 Chapters)
- Analytic Number Theory: Advanced Topics
- Sieve Methods
- The Circle Method
- The Large Sieve
- Exponential Sums
- L-functions: General Theory
- The Functional Equation of the Zeta Function
- The Distribution of Primes: Advanced Topics
- Prime Number Theorem for Arithmetic Progressions
- The Chebotarev Density Theorem
- Algebraic Number Theory: Advanced Topics
- Galois Theory and Number Fields
- Class Field Theory: Introduction
- Local Fields and Global Fields
- Adeles and Ideles
- Arithmetic Geometry: Introduction
- Elliptic Curves: Advanced Topics
- Modular Forms: Advanced Topics
- The Weil Conjectures (Statement)
- The Sato-Tate Conjecture
- Diophantine Equations: Advanced Topics
- The Mordell-Weil Theorem
- Heights and the Canonical Height
- The Thue-Siegel Theorem
- Transcendental Number Theory: Advanced Topics
- Gelfond-Schneider Theorem
- Transcendence Methods
- Diophantine Approximation: Advanced Topics
- The Subspace Theorem
- Applications of Number Theory in Cryptography
IV. Further Explorations and Specialized Topics (20 Chapters)
- Computational Number Theory: Algorithms
- Primality Testing Algorithms
- Integer Factorization Algorithms
- Elliptic Curve Cryptography: Implementation
- Quantum Computing and Number Theory
- Number Theory and Coding Theory
- Number Theory and Combinatorics
- Number Theory and Dynamical Systems
- Number Theory and Physics
- Number Theory and Music
- The History of Number Theory
- Famous Problems in Number Theory
- Open Problems in Number Theory
- The Riemann Hypothesis: Current Status
- The Langlands Program: Introduction
- Modular Arithmetic and Computer Arithmetic
- Number Theory and Machine Learning
- Number Theory and Data Science
- Connections Between Number Theory and Other Fields
- Appendix: Foundational Material and References