Primitive Roots

Let a and n be relatively prime positive integers. The smallest positive integer k so that a^k ≡ 1 (mod n) is called the order of a modulo n. The order of a modulo n is usually denoted by ord_na. If it happens that ord_na = f(n), where f is the Euler phi function, then a is said to be a primitive root modulo n.

It is well known which integers have primitive roots. It is also known that if the positive integer n has a primitive root, then it has a total of f(f(n)) incongruent primitive roots. The purpose of this lab is to attempt to discover which integers have primitive roots.

Exercises

Zn Applet Full Screen Help System for Single Group Viewers

1. Use the ring structure of Z_n and the applet to the right to find a primitive root modulo each of the following integers.
   1. 8
   2. 14
   3. 11
   4. 25
   5. 26
   6. 6
2. Show that the integer 15 has no primitive roots.
3. Show that the integer 20 has no primitive roots.
4. How many incongruent roots does 11 have? Find a set of this many incongruent primitive roots modulo 11.
5. Make a conjecture about which positive integers n possess a primitive root.
6. Find all solutions of the congruences (express your answer as a congruence class).
   1. 7^k = 13 (mod 15)
   2. 5^k = 4 (mod 19)