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

5x^{2} + 11y^{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 = 4q + 1 where q is a prime, then 2 must be primitive mod p.

Document network location:

DVI: http://math.albany.edu:8000/math/pers/hammond/course/mat4525/assgt/nt000211.dvi
HTML: http://math.albany.edu:8000/math/pers/hammond/course/mat4525/assgt/nt000211.html