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**Unformatted text preview: **n cannot be represented as the sum of two squares of rational numbers. (5) Prove that the positive integer n has as many representations as the sum of two squares as does the integer 2 n . [Hint: Starting with a representation for n as a sum of two squares, obtain a similar representation for 2 n , and conversely.] (6) Prove that of any four consecutive integers, at least one of them is not representable as a sum of two squares. (7) For any n > 0, show that there is a positive integer that can be expressed in n distinct ways as a sum of two squares. [Hint: Note that, for k = 1 , 2 , . . . , n , 2 2 n +1 = (2 2 n-k + 2 k-1 ) 2-(2 2 n-k-2 k-1 ) 2 . ] 1...

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