hw_04

Hw_04 - Exercise 3 2.8.22 Let p be prime and let g be a primitive root mod p Show that p-1 g g 2 ·· g p-1 mod p and give another proof of

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Number Theory, Homework 3rd batch due: Wednesday, February 18 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. Sometimes, only five problems will be assigned, in which case the selection issue is moot. Exercise 1. 2.8.16: Let m and n be positive integers with m being odd. Show that 2 m - 1 and 2 n + 1 are relatively prime. Exercise 2. 2.8.17: Assume that a > 1 and that a k + 1 is prime. Show that k is a power of 2.
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Unformatted text preview: Exercise 3. 2.8.22: Let p be prime and let g be a primitive root mod p . Show that ( p-1)! ≡ g · g 2 ··· g p-1 mod p and give another proof of Wilson’s congruence (Theorem 2.11). Exercise 4. 2.8.35: Let p be prime. Show that there are infinitely many primes congruent 1 mod p . Exercise 5. 2.8.37: Assume n > 1. Show that n does not divide 2 n-1. Exercise 6. 2.8.22: Let p be prime and let g be a primitive root mod p . Show that ( p-1)! ≡ g · g 2 ··· g p-1 mod p and give another proof of Wilson’s congruence (Theorem 2.11)....
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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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