th_01

# th_01 - that everybody is on the same page Exercise 1 For a...

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Number Theory 1st Take Home due: Wednesday, March 25 All exercises are worth 5 points. Please select ﬁve problems since there is a cap of 25 points. If you decide to do more than ﬁve problems, please indicate which ones you want graded since otherwise the grader will be free to chose problems at will. Hint: Not all problems are equally hard, so ﬁnding the ones not to submit is part of the task. This is a take home. It’s just like home work, except that it is (a) comprehensive, (b) closed book and (c) to be done without collaboration; the last point includes me: I won’t answer questions on the assignment during oﬃce hours. Instead, if you feel that a problem needs clariﬁcation, send me an e-mail. I will answer to the class, so
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Unformatted text preview: that everybody is on the same page. Exercise 1. For a, b > 2, show that 2 a + 1 is not divisible by 2 b-1. Exercise 2. Let n > d > 0 and assume that d divides n . Show that n-φ ( n ) > d-φ ( d ) . Exercise 3. Let p be a prime and assume that a has order 3 in the multiplicative group Z * p . Show that (1 + a ) has order 6. Exercise 4. Evaluate the Jacobi symbol ( 103 111 ) . Exercise 5. Let p be a prime congruent 1 mod 4 and assume a 2 + b 2 = p where a is odd. Show that ‡ a p · = 1 . Exercise 6. Show that 1 φ ( n ) = 1 n X d | n μ ( d ) 2 φ ( d ) for all positive integers n . Exercise 7. Show that ∑ ∞ n =1 d ( n ) 2 n s = ζ ( n ) 4 ζ (2 s ) ....
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## This note was uploaded on 04/05/2010 for the course CS 150 taught by Professor Humphreys,g during the Spring '08 term at UVA.

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