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# hw9 - s and t are both odd and s t = 1 then s-t s t =...

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Math 465 Problem Set 9 Due Friday, April 25, 2008 1. Prove that { 3 , 4 , 5 } is the only primitive Pythagorean triple involving consecutive pos- itive integers. 2. Find all Pythagorean triangles whose areas are equal to their perimeters. (Hint: Solve x 2 + y 2 = z 2 and xy = 2( x + y + z ).) 3. Prove that the equation x 2 + y 2 = z 3 has inﬁnitely many solutions for x, y, z positive integers. (Hint: For any n > 1, let x = n ( n 2 - 3) and y = 3 n 2 - 1.) 4. The equation x 4 - y 4 = 2 z 2 has no solutions in positive integers x, y, z . Prove it following the suggested steps. Step 1. Suppose that the equation x 4 - y 4 = 2 z 2 is solvable. Step 2. Show that x, y must be both odd or both even. Step 3. Show that if
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Unformatted text preview: s and t are both odd and ( s, t ) = 1 then ( s-t, 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 , x-y = 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 , x-y = 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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