Prob3 - 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

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MATH 115B PROBLEM SET III APRIL 24, 2007 (1) Establish each of the following assertions: (a) Each of the integers 2 n where n = 1 , 2 , . . . , is a sum of two squares. (b) If n 3 or 6 (mod 9), then n cannot be represented as a sum of two squares. (c) Every Fermat number F n = 2 2 n + 1, where n 1 can be expressed as a sum of two squares. (2) Show that a positive integer n is a sum of two squares if and only if n = 2 m a 2 b , where m 0, a is an odd integer, and every prime divisor of b is of the form 4 k + 1. (3) Find a positive integer having at least three different representations as the sum of two squares. [Hint: Choose an integer that has three distict prime factors, each of the form 4 k + 1.] (4) If the positive integer n is not the sum of two squares of integers, show that
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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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