Math 58 - Number Theory

October 26, 2006

Homework 11? (due October 31):

About primitive roots:

1. Find all of the primitive roots (= generators of Z_m^*) mod 13.

2. Find all of the primitive roots mod 17.

3. If p = 1 (mod 4) and p is prime, and r is a primitive root mod p, show that (-r) is also a primitive root mod p.

About quadratic residues:

4. If p is prime, and a and b are squares (= quadratic residues) mod p, and c and d are not squares mod p (and if none of a, b, c, d is zero mod p) then show that…

ab is a square

ac is not a square

cd is a square, all mod p.

5. Show that 4 and 9 are never primitive roots mod p, for any prime p.

*6. (This is problem 28 on p. 414 of the text, where there’s a hint. This probably requires reading Theorem 11.6 in the text.) Show that there are infinitely many primes of

each of these forms…

= 3 mod 8

= 5 mod 8

= 7 mod 8

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