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MATH135_F11_Assignment_8_Solns - 1 MATH 135 F 2011...

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1 MATH 135 F 2011: Assignment 8 Due: 8:30 AM, Wednesday, 2011 Nov. 23 in the dropboxes outside MC 4066 Write your answers in the space provided. If you wish to typeset your solutions, use the solution template posted on the course web site. Cite any proposition or definition you use. Family Name: First Name: I.D. Number: Section: Mark: (For the marker only.) If you used any references beyond the course text and lectures (such as other texts, discussions with colleagues or online resources), indicate this information in the space below. If you did not use any aids, state this in the space provided. In this assignment, you will complete a proof of Fermat’s Last Theorem for the case n = 3. Theorem 1 (FLT 3) . The Diophantine equation x 3 + y 3 = z 3 has no positive integer solution. The proof is on the next page and follows a proof by Euler as described in http://fermatslasttheorem. blogspot.com/2005/05/fermats-last-theorem-proof-for-n3.html . You may consult this web site but all of your work must be expressed in a manner consistent with the presentation below and propositions proved in class. The proof makes use of the following propositions. Proposition 1. If x , y and z are integers, not all zero, and gcd( x, y ) = 1 , then gcd( x, y, z ) = 1 . Proposition 2. If m is odd, m | ( a 2 + 3 b 2 ) and gcd( a, b ) = 1 , then there exist integers c and d so that m = c 2 + 3 d 2 . You may assume that all of the values that appear in this proof are integers.
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2 Proof. The proof of FLT 3 will proceed in parts. You will be asked questions about some of the parts. Part 1: By way of contradiction, suppose that that there exists a solution to x 3 + y 3 = z 3 . Of all such solutions, choose one in which z is as small as possible. Call this solution
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MATH135_F11_Assignment_8_Solns - 1 MATH 135 F 2011...

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