Number Theory

  1. Diophantine Equations
    1. Introduction to Diophantine Equations
      1. Definition and Classification
        1. Historical Background
          1. Types of Diophantine Equations
          2. Linear Diophantine Equations
            1. Two Variables
              1. Form ax + by = c
                1. Solvability Conditions
                  1. Finding Particular Solutions
                    1. General Solution Structure
                    2. Multiple Variables
                      1. General Linear Equations
                        1. Solution Methods
                      2. Pythagorean Triples
                        1. Definition and Basic Properties
                          1. Primitive Pythagorean Triples
                            1. Parametric Solutions
                              1. Euclid's Formula
                                1. Proof of Completeness
                                2. Applications and Extensions
                                3. Pell's Equation
                                  1. Standard Pell Equation x² - Dy² = 1
                                    1. Fundamental Solution
                                      1. General Solutions
                                        1. Negative Pell Equation
                                          1. Connection to Continued Fractions
                                          2. Sum of Squares Problems
                                            1. Sum of Two Squares
                                              1. Characterization Theorem
                                                1. Proof Techniques
                                                2. Sum of Three Squares
                                                  1. Legendre's Three-Square Theorem
                                                  2. Sum of Four Squares
                                                    1. Lagrange's Four-Square Theorem
                                                      1. Jacobi's Four-Square Theorem
                                                    2. Fermat's Last Theorem
                                                      1. Statement and History
                                                        1. Special Cases
                                                          1. Modern Proof Outline
                                                            1. Impact on Mathematics

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                                                          7. Arithmetic Functions

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                                                          9. Quadratic Residues and Reciprocity