A question that is related to Research Question 1 is the following:
How many square roots mod p can an integer have?
We say that m is a square root of
a modulo p if a 
m2 mod p. The applet below returns a list
of all square roots of a mod p.
For example, here are the square roots of 3 modulo 11:
You can check the results using the Java calculator:
Research Question 2Let p be an odd prime and a an integer that is not divisible by p. How many square roots can a have modulo p? |
Once you have completed Research Question 2, you may find that you have enough information to prove your conjecture for Research Question 1. Go back, and give it a try. If you have success, great! If not, don't despair. Go on to the next section, and return to Research Question 1 later.