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Homework 10

# Homework 10 - x and y satisfying x 2 y 2 = n 5 Give an...

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MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28) 1(a). Problem 15.2 on page 198 of Eccles. 1(b). Give an example of an integer n with the property that the statement of 1(a), with n substituted for 5, becomes a false statement. Give an explicit example of an integer a for which one of the conditions holds but the other doesn’t. 1(c). Give a few more examples of numbers like 5 (for which the statement of 1(a) is true with that number in place of 5), and a few more numbers like your n (for which the statement of 1(a) is false with that number in place of 5). You do not need to prove anything, just give examples. 2. Problem 15.3 on page 198 of Eccles. 3. Problem 15.5 on page 198 of Eccles. (The first sentence makes enough sense as it is. For “Deduce that 1234567 is not a perfect square,” I would read “Explain, using the division theorem, why this means that 1234567 is not a perfect square.”) 4. Suppose that n is a positive integer, and that 3 | n , but that 9 6 | n . Prove that there are no integers
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Unformatted text preview: x and y satisfying x 2 + y 2 = n. 5. Give an example of a positive integer n such that 9 | n and such that there are integers x and y with the property that x 2 + y 2 = n. (Give an example of such an x and y .) 6(a). Suppose m is a positive integer with the property that there are no positive integers x and y satisfying x 2 + y 2 = m. Prove that there cannot be any positive integers x and y satisfying x 2 + y 2 = 9 m. 6(b). Give an example of a number k such that 9 | k and yet there are no integers x and y with the property that x 2 + y 2 = k . 7(a). Prove (any way you like) that there are no integers x,y,z satisfying x 2 + y 2 + z 2 = 7 . 7(b). Prove that if m has the property that there are no integer solutions to x 2 + y 2 + z 2 = m, then there are also no integer solutions to x 2 + y 2 + z 2 = 4 m. Give a formula that describes an inﬁnite set of integers that cannot be written as a sum of three integer squares....
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