2020 IB Extended Essays

3.3 Reflection on 3.0 Pell’s Equation The objective of 3.0 Pell’s Equation was to research, understand, and summarize the algorithm for obtaining solutions to Pell’s equations. I believe I have sufficiently met this objective, understanding to considerable depth the mathematical principles underlying Pell’s equation, using a variety of resources that I judge were sufficient and reliable. By compiling, understanding, analyzing, condensing, and reenacting the algorithmic process for arriving at the solutions to Pell’s equation, I was able to genuinely appreciate and internalize the multifaceted approach to the problem. However, the investigation reveals several shortcomings throughout. At quite a few stages, I fail to provide direct proof of the discoveries I make from observation or to provide exhaustive explanation for the mathematical concepts which are assumed without justification. For example, I discussed the eventually periodic and palindromic properties of continued fraction notations, but I did not go on to prove these properties or to prove these properties manifest in all such square root continued fraction notations. Furthermore, although I demonstrated that the solutions to Pell’s equation can be found in a portion of the continued fraction notation, I did not then further prove that they cover all solutions—that there are no other solutions than the ones found in the continued fraction notation. In addition to these kinds of further proofs that are naturally prompted for a more thorough and complete establishment of the Pell’s equation algorithm, there were several sources of potential exploration left uninvestigated. For example, in 3.2.4 Generating All Solutions to Pell’s Equation from the Fundamental Solution , only the simplistic, most straight-forward method for generating new solutions from given ones is investigated. However, as I came across in my preparatory research, there exists another, perhaps more stimulating, method proposed by Brahmagupta, an Indian mathematician from the 7 th century. Not only does

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