Unformatted text preview: s and t are both odd and ( s, t ) = 1 then ( st, s + t ) = 2. (Hint: Refer to Question 2 from HW set 2.) Step 4. If x, y are both odd, assume ( x, y ) = 1 and show that x 2 + y 2 = 2 a 2 , x + y = 2 b 2 , xy = 2 c 2 for some a, b, c ; hence, a 2 = b 4 + c 4 . Step 5. If x, y are both even, assume x = 2 x and y = 2 y with ( x , y ) = 1. Use Step 3 to show that x 2 + y 2 = 8 a 2 , x + y = 4 b 2 , xy = 4 c 2 for some a, b, c ; hence, a 2 = b 4 + c 4 . Step 6. Use the theorem that the equation x 4 + y 4 = z 2 has no solutions. 1...
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 Spring '08
 YEE
 Number Theory, Angles, Pythagorean Theorem, Integers, Euclidean algorithm, positive integers

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