It is easy to have Java produce a list of quadratic residues by automating the method used in the Prelab section. Specifically, the applet below computes

a^{2}%nfor each value ofasatisfying gcd(a,n) = 1 and 1an. The resulting output contains the quadratic residues. Try it out:

In the very near future, you will want to count the number of quadratic residues modulo a prime. You can do so using a modified version of the previous applet that includes the number of quadratic residues in the list, thus avoiding the tedium of counting them yourself.

## Research Question 1

If

pis an odd prime, how many quadratic residues are there modp?

Hint:The content of the next few sections my be helpful in proving your conjecture.

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

Copyright © 2001 by W. H. Freeman and Company