2020 IB Extended Essays

7.0 Conclusion

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

Overall, this essay has effectively and thoroughly examined the solution and proof for Pell’s equation and Fermat’s Sandwich Theorem, with the appropriate background information provided and coherent logical steps. Not only that, the assumptions made in the process that lacked sufficient proof were also clearly identified and raised possible amendments for future investigations to. With due consideration of the several identified shortcomings and accordingly suggested modifications in the evaluations above, the topics of Pell’s equation and Fermat’s Sandwich Theorem discussed and analysed in this essay can be contextualized in their respective broader mathematical settings. Pell’s equation 2 − 2 = 1 , especially in light of the magic tables used to visualize the problem, can be used to find units in quadratic rings and prove results from those, to identify simultaneously polygonal (e.g. simultaneously triangular and square, such as 1 = 1 × 2 2 = 1 2 , 36 = 8 × 9 2 = 16 2 , 1225 = 49 × 50 2 = 35 2 , etc.) numbers, to approximate √ as a rational value (e.g. √ 7 ≈ 717 271 and √ 13 ≈ 4287 1189 , etc.), etc. As for Fermat’s Sandwich Theorem, the defining equation of the problem— 3 = 2 + 2 —is a specific case of the Mordell curve— 3 = 2 + , where is a non-zero integer constant—, which again is a specific kind of elliptic curve—a non-singular plane algebraic curve defined by the equation 3 = 3 + + .

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