2020 IB Extended Essays

3. Therefore, the only integer solutions , to 3 = 2 + 2 are = ±5, = 3 ; 26 is the only integer sandwiched between a perfect square number ( 25 = 5 2 ) and a perfect cubic one ( 27 = 3 3 ). 4.4 Reflection on 4.0 Fermat’s Sandwich Theorem: 3 = 2 + 2 The objective of 4.0 Fermat’s Sandwich Theorem: 3 = 2 + 2 was to research, understand, and summarize the proof for Fermat’s Sandwich Theorem. I conclude I have comprehended, to considerable depth, the proof for Fermat’s Sandwich Theorem by systematically explaining and analyzing the Fermat’s Sandwich Theorem and the underlying mathematical framework, based on sufficient and reliable resources. However, the investigation comes short at a few stages throughout, where I fail to provide sufficient justification for certain mathematical facts. For example, I take it as given that the ring ℤ�√− 2 � is a UFD, that 2 is a Heegner number, meaning all the algebraic integers in the ring ℤ�√− 2 � are uniquely factorized. Another fact simply accepted without due proof is that rings ℤ / 8ℤ and ℤ are isomorphic and that, therefore, the modulo relationship that holds in congruence ring ℤ / 8ℤ also holds in ℤ . Despite the fact that the two major assumptions are central for the proof to be continuous and complete, addressing both to satisfaction would have exceeded the scope and depth manageable currently in this essay. Not only were the two naturally prompted leads above not followed, but no alternative interpretations or approaches to the problem were explored or considered, either. Although the different resources I consulted in my research differed in their explanations for certain parts of the proof—which I revised, edited, and synthesized to best suit my understanding and to most straightforwardly lay out the crucial points—, the gist of their approaches and the area of mathematics involved were virtually the same.

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