MATH 319 – Class Exercise on Power Residues and Primitive Roots
Names______________________________________________________________
1.
2
is a primitive root mod 19. Use the
following short Maple program to fill in the table of powers of 2 mod 19.
Program
A
for k from 1 to 18 do
k,modp(2^k,19);
od;
|
n |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
2n(mod19) |
2 |
|
|
|
|
|
|
|
|
|
n |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
2n(mod19) |
|
|
|
|
|
|
|
|
|
2.
The
next program will square each number in the set {1,2,3…18} and reduce the
square mod 19. This will result in a
list of the quadratic residues mod 19.
Program
B
for k from 1 to 18 do
k,modp(k^2,19);
od;
|
x |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
x2mod19 |
|
|
|
|
|
|
|
|
|
|
x |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
x2mod19 |
|
|
|
|
|
|
|
|
|
3. Return to the table in part one and circle the powers of 2 that are quadratic residues. Which powers do the quadratic residues correspond to?____________________
4.
Theorem Let p be a prime and (a,p) = 1. Then the congruence 
has
either (n,p-1)
or no solutions. It has (n,p-1)
solutions if 
and no solutions otherwise.
Using
this theorem, answer the following questions.
How many solutions are there to 
? ________.
What
are the solutions? ___________.
Check
the congruence condition at the end of the theorem. That is, compute
(p-1)/(n,p-1) and raise 6 to that power (mod 19). _________________
5. Alter Program B to find all the cubic (power
three) residues mod 19.
Cubic Residues (mod 19)
_______________________________
Check
the powers of 2 again. To which powers
do the cubic residues correspond? __________________________
8 is a cubic residue. Using the theorem, how many solutions are
there to 
? ________ What are
the solutions? ____________