MATH_135_21_FLT4_Beamer_Student - Objectives Fermats Last Theorem Proof Reducing FLT History MATH 135 Faculty of Mathematics University of Waterloo

MATH_135_21_FLT4_Beamer_Student - Objectives Fermats Last...

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Objectives Fermat’s Last Theorem - Proof Reducing FLT History MATH 135 Faculty of Mathematics, University of Waterloo Lecture 21: Fermat’s Theorem for n = 4 Faculty of Mathematics, University of Waterloo MATH 135
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Objectives Fermat’s Last Theorem - Proof Reducing FLT History Content Content 1. State and prove: The Diophantine equation x 4 + y 4 = z 2 has no positive integer solution. 2. State and prove: The Diophantine equation x 4 + y 4 = z 4 has no positive integer solution. 3. Show reduction of FLT to If p is an odd prime, then the Diophantine equation x p + y p = z p has no positive integer solution. Faculty of Mathematics, University of Waterloo MATH 135
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Objectives Fermat’s Last Theorem - Proof Reducing FLT History 1. Proof By Contradiction 2. Finding a Primitive Pythagorean Triple 3. What DosandtLook Like? 4. Finding Another Primitive Pythagorean Triple 5. The Contradiction FLT, Strong Version of n = 4 Theorem (FLT, Strong Version of n = 4) The Diophantine equation x 4 + y 4 = z 2 has no positive integer solution. Faculty of Mathematics, University of Waterloo MATH 135
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Objectives Fermat’s Last Theorem - Proof Reducing FLT History 1. Proof By Contradiction 2. Finding a Primitive Pythagorean Triple 3. What DosandtLook Like? 4. Finding Another Primitive Pythagorean Triple 5. The Contradiction Structure 1. Proof by contradiction 2. Finding a primitive Pythagorean triple 3.What dosandtlook like? 4. Finding another primitive Pythagorean triple 5. The contradiction Faculty of Mathematics, University of Waterloo MATH 135
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Objectives Fermat’s Last Theorem - Proof Reducing FLT History 1. Proof By Contradiction 2. Finding a Primitive Pythagorean Triple 3. What DosandtLook Like? 4. Finding Another Primitive Pythagorean Triple 5. The Contradiction 1. Proof By Contradiction Suppose that there does exist a positive integer solution to x 4 + y 4 = z 2 . Of all such solutions x 0 , y 0 , z 0 , choose any one in which z 0 is smallest. Without loss of generality, we may also assume that gcd( x 0 , y 0 ) = 1. (Why?) This in turn implies that gcd( x 0 , y 0 , z 0 ) = 1. (Why?) Note: The last sentences of the proof are: But recall that x 0 , y 0 , z 0 is a solution to x 4 + y 4 = z 2 with the smallest possible value of z . But x 1 , y 1 , z 1 is a solution to x 4 1 + y 4 1 = z 2 1 with a smaller value of z ! Faculty of Mathematics, University of Waterloo MATH 135
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Objectives Fermat’s Last Theorem - Proof Reducing FLT History 1. Proof By Contradiction 2. Finding a Primitive Pythagorean Triple
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