## 11.3   Counting Square Roots

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:

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You can check the results using the Java calculator:

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#### Research Question 2

Let 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.

Section 11.1 | Section 11.2 | Section 11.3 | Section 11.4 | Section 11.5 | Section 11.6