Number Theory Assignment

due Friday, February 18, 2000

1. What is the smallest positive primitive root modulo the prime power 5^{100} ? Justify your answer without a great deal of computation.
   
   
   

2. If p is an odd prime, then 2 cannot be primitive when it is a square mod p. The first instance of this is the prime p = 7. Find the smallest odd prime p for which 2 is not primitive mod p and for which 2 is not a square mod p.
   
   
   

3. Can you find the smallest integer m >= 2 with the property that there are no integers x, y for which

   5 x^{2} + 11 y^{2} = 1 (mod m) ?

   
   
   
   

4. Find all solutions

   c_{0} + c_{1} 7 + c_{2} 7^{2} + c_{3} 7^{3} + c_{4} 7^{4} + c_{5} 7^{5} with 0 <= c_{j} < 7

   of the congruence

   x^{3} - x + 1 = 0 (mod 7^{6}) .

   
   
   
   

5. Prove that if p is a prime of the form p = 4 q + 1 where q is a prime, then 2 must be primitive mod p.

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