2020 IB Extended Essays

4.0 Fermat’s Sandwich Theorem: = + Fermat’s Sandwich Theorem states that 26 is the only number sandwiched between a perfect square number ( 25 = 5 2 ) and a perfect cubic one ( 27 = 3 3 ). In other words, the only solutions to the Diophantine equation 3 = 2 + 2 are = ±5, = 3 . According to Simon Singh, author of Fermat’s Last Theorem , Fermat challenged other mathematicians to achieve this result without revealing his own proof, taking delight in taunting the English mathematicians Wallis and Digby with their inability to prove the result (Weisstein, Eric W [unknown]). 4.1 Unique Factorization in Imaginary Quadratic Ring ℤ�√− 2 � Consider the quadratic ring ℤ�√ � . If > 0 , the ring is a real quadratic ring, and if < 0 , the ring is an imaginary quadratic ring. In mathematics, a unique factorization domain (UFD) is an integral domain—a commutative ring with no zero divisors—in which every non-zero, non-unit element can be expressed as a unique product of irreducible elements. Irreducible element ∈ cannot be written as = where neither nor are non-units. Not all imaginary quadratic rings are uniquely factorable; the algebraic integers in the some imaginary quadratic rings do not necessarily have unique factorizations. However, the ring ℤ�√− 2 � is a UFD, because 2 is a Heegner number, meaning that the algebraic integers in the ring ℤ�√− 2 � have unique factorization. For example, 3 = 1 ∙ 3 = (1 + √− 2)(1 − √− 2) All other quadratic rings ℤ�√ � where �√ � ≤ 7 are uniquely factorable (Weisstein, Eric W [unknown]).

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