2020 IB Extended Essays

1.0 Introduction

Research Question: How can the solutions to Pell’s Equation be acquired, and how can Fermat’s Sandwich Theorem be proved?

The initial inspiration from Simon Singh’s Fermat’s Last Theorem (2004) developed and had me exploring two-variable linear Diophantine equations and three methods of generating primitive Pythagorean triples in my HL Maths IA, where I came across Pell’s equation. Eventually, I arrived at this research question in order to understand and summarize the method for finding solutions to Pell’s equations and in order to reconstruct the proof for Fermat’s Sandwich Theorem. Both Pell’s equation and Fermat’s Sandwich Theorem are quintessential problems in the field of number theory, a branch of pure mathematics primarily concerned with integers and sometimes called “higher arithmetic” for being among the oldest and most natural of mathematical pursuits (Dunham, W [unknown]). Primes and prime factorization are vital concepts in number theory, which will be explained and incorporated into this investigation of both Pell’s equation and Fermat’s Sandwich Theorem (Weisstein, Eric W [unknown]). Until the mid-20 th century, number theory was considered to be the purest of mathematics, leading to no direct applications in the real life. However, the emergence of digital computers and communications uncovered a strong mutuality between the seemingly remote academic fields of number theory and computer technology. Number theory is helping answer real- world problems and computer technology is assisting number theorists factor large numbers, determine primes, test conjectures, and solve problems previously deemed unapproachable (Dunham, W [unknown]).

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