Exploring the Foundations of Number Theory: A Historical and Mathematical Investigation

1. Introduction 1.1 Background of Number Theory 1.2 Objectives of the Study 1.3 Scope and Limitations 1.4 Structure of the Paper 2. Historical Overview 2.1 Early Number System Concepts 2.2 Ancient Civilizations' Contributions 2.3 Medieval Advances in Number Theory 3. Fundamental Theorems 3.1 Euclidean Algorithm 3.2 Fundamental Theorem of Arithmetic 3.3 Fermat's Little Theorem 4. Prime Numbers 4.1 Definition and Properties 4.2 Sieve of Eratosthenes 4.3 Distribution and Patterns 5. Diophantine Equations 5.1 Introduction to Diophantine Problems 5.2 Linear Diophantine Equations 5.3 Non-linear Diophantine Equations 6. Algebraic Number Theory 6.1 Introduction and Basic Definitions 6.2 Quadratic Fields 6.3 Cyclotomic Fields 7. Modern Applications 7.1 Cryptography 7.2 Computational Number Theory 7.3 Number Theory in Physics 8. Conclusion 8.1 Summary of Findings 8.2 Future Research Directions 8.3 Final Thoughts and Reflections