Integers, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods.
This volume studies integers and arithmetic structures. It covers divisibility, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods.
Part I. Integers and Divisibility
Chapter 1. Basic Properties of Integers
1.1 Divisibility and factors 1.2 Prime numbers 1.3 Greatest common divisor 1.4 Euclidean algorithm 1.5 Fundamental theorem of arithmetic
Chapter 2. Arithmetic Functions
2.1 Definition and examples 2.2 Multiplicative functions 2.3 Möbius function 2.4 Euler totient function 2.5 Dirichlet convolution
Chapter 3. Congruences
3.1 Modular arithmetic 3.2 Residue classes 3.3 Linear congruences 3.4 Chinese remainder theorem 3.5 Applications
Part II. Diophantine Equations
Chapter 4. Linear Diophantine Equations
4.1 Existence of solutions 4.2 General solutions 4.3 Applications 4.4 Systems of equations 4.5 Integer constraints
Chapter 5. Quadratic Equations
5.1 Sum of squares 5.2 Pell equations 5.3 Quadratic forms 5.4 Representations of integers 5.5 Methods of solution
Chapter 6. Higher-Degree Equations
6.1 Polynomial equations over integers 6.2 Rational solutions 6.3 Elliptic curves (overview) 6.4 Diophantine approximation 6.5 Famous problems
Part III. Analytic Number Theory
Chapter 7. Distribution of Primes
7.1 Prime counting function 7.2 Chebyshev estimates 7.3 Prime number theorem (overview) 7.4 Error terms 7.5 Applications
Chapter 8. Dirichlet Series
8.1 Definitions and convergence 8.2 Riemann zeta function 8.3 Euler products 8.4 Analytic continuation (overview) 8.5 Applications
Chapter 9. Additive Number Theory
9.1 Partition problems 9.2 Goldbach-type problems 9.3 Waring’s problem 9.4 Circle method (overview) 9.5 Applications
Part IV. Algebraic Number Theory
Chapter 10. Algebraic Numbers
10.1 Algebraic integers 10.2 Minimal polynomials 10.3 Field extensions 10.4 Conjugates and norms 10.5 Examples
Chapter 11. Number Fields
11.1 Ring of integers 11.2 Ideals and factorization 11.3 Unique factorization issues 11.4 Class groups 11.5 Units and structure
Chapter 12. Local Fields
12.1 p-adic numbers 12.2 Valuations 12.3 Completions 12.4 Local-global principles 12.5 Applications
Part V. Modular Forms and Advanced Topics
Chapter 13. Modular Arithmetic Structures
13.1 Multiplicative groups modulo n 13.2 Primitive roots 13.3 Quadratic residues 13.4 Reciprocity laws (overview) 13.5 Applications
Chapter 14. Modular Forms (Overview)
14.1 Basic definitions 14.2 Fourier expansions 14.3 Hecke operators 14.4 Connections to number theory 14.5 Applications
Chapter 15. Elliptic Curves (Overview)
15.1 Group law 15.2 Rational points 15.3 Reduction modulo primes 15.4 Applications in cryptography 15.5 Links to Diophantine problems
Part VI. Computational Number Theory
Chapter 16. Algorithms on Integers
16.1 Fast arithmetic 16.2 GCD algorithms 16.3 Modular exponentiation 16.4 Primality testing 16.5 Factorization algorithms
Chapter 17. Cryptographic Applications
17.1 Public-key cryptography 17.2 RSA and variants 17.3 Discrete logarithm problem 17.4 Elliptic curve cryptography 17.5 Security assumptions
Chapter 18. Experimental Number Theory
18.1 Computational exploration 18.2 Data-driven conjectures 18.3 Verification of results 18.4 High-performance methods 18.5 Examples
Part VII. Geometric and Topological Methods
Chapter 19. Geometry of Numbers
19.1 Lattices in Euclidean space 19.2 Minkowski’s theorem 19.3 Convex bodies 19.4 Applications 19.5 Counting lattice points
Chapter 20. Arithmetic Geometry (Overview)
20.1 Varieties over number fields 20.2 Rational points 20.3 Local-global principles 20.4 Heights and measures 20.5 Applications
Chapter 21. Topological Methods
21.1 Cohomology ideas (overview) 21.2 Étale methods (overview) 21.3 Arithmetic topology analogies 21.4 Connections to geometry 21.5 Research directions
Part VIII. Open Problems and Research
Chapter 22. Major Conjectures
22.1 Riemann hypothesis 22.2 Birch–Swinnerton-Dyer (overview) 22.3 Goldbach conjecture 22.4 Twin prime conjecture 22.5 Other open problems
Chapter 23. Interdisciplinary Connections
23.1 Number theory and physics 23.2 Number theory and coding 23.3 Number theory and combinatorics 23.4 Number theory and computer science 23.5 Emerging areas
Chapter 24. Future Directions
24.1 Computational expansion 24.2 Data-driven mathematics 24.3 Hybrid methods 24.4 Large-scale collaboration 24.5 Long-term outlook
Appendix
A. Common number-theoretic functions B. Key theorems summary C. Algorithm templates D. Computational tools E. Cross-reference to other MSC branches
This volume develops number theory from elementary arithmetic to modern research directions, balancing classical proofs with computational and structural methods.