th_01

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

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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- ( n ) > d- ( 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 ( n ) = 1 n X d | n ( d ) 2 ( d ) for all positive integers n . Exercise 7. Show that n =1 d ( n ) 2 n s = ( n ) 4 (2 s ) ....
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