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781hw4 - Write a PARI program to test Artin’s conjecture...

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18.781, Fall 2007 Problem Set 4 Due: FRIDAY, October 5 1. Complete the following problems from Niven-Zuckerman-Montgomery (henceforth NZM): NZM 2.7: 2, 3, 4 NZM 2.8: 2, 6, 8, 9, 14, 18, 27, 31, 35 2. PARI PROGRAM OF THE WEEK: A famous conjecture of (Emil) Artin states: There are infinitely many primes p for which 2 is a primitive root mod p . Recall a number a mod m is called a primitive root mod m if the order of a mod m is φ ( m ).
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Unformatted text preview: Write a PARI program to test Artin’s conjecture. Try to determine a function whose growth matches that of P 2 ( X ), the number of primes p ≤ X for which 2 is a primitive root mod p (in the same spirit as our earlier calculations for primes in arithmetic progressions). Can you explain why the function you guessed makes sense as a model for the growth of P 2 ( X )?...
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