2020 IB Extended Essays

Table of Contents

1.0 Introduction . . . page 3 2.0 Quadratic Rings . . . page 5

2.1 Algebraic Integers and Quadratic Rings . . . page 5

3.0 Pell’s Equation . . . page 6

3.1 Fundamental Observations . . . page 6 3.2 Dirichlet’s Unit Theorem and Continued Fraction Expansion of √ . . . page 7 3.2.1 Continued Fraction Notation . . . page 8 3.2.2 The “Magic Table” Algorithm . . . page 10 3.2.3 Observations of the Magic Table for √ and the Solutions to 2 − 2 = 1 . . . page 13 3.2.4 Generating All Solutions to Pell’s Equation from the Fundamental Solution . . . page 14 3.3 Reflection on 3.0 Pell’s Equation . . . page 16 4.0 Fermat’s Sandwich Theorem: 3 = 2 + 2 . . . page 18 4.1 Unique Factorization in Imaginary Quadratic Ring ℤ�√− 2 � . . . page 18 4.2 Quadratic Residues and Relative Primality of + √ 2 and − √ 2 . . . page 19 4.3 ± √ 2 as a Cube in ℤ�√− 2 � . . . page 20 4.4 Reflection on 4.0 Fermat’s Sandwich Theorem: 3 = 2 + 2 . . . page 21

5.0 Conclusion . . . page 23 6.0 Bibliography . . . page 25

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