Solution: Exercise 1. This software finds all solutions to

Chapter 12, Problem 6CE

(choose chapter or problem)

This software determines the number of solutions to the equation \(X^2 = -I\) where X is a 2 x 2 matrix with entries from \(Z_n\) and I is the identity. Run the program for n = 32 . Make a conjecture about the number of solutions when n = 2 k where k > 1 . Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3 . Run the program for n = 27 and 49 . Make a conjecture about the number of solutions when n has the form \(p^i\) where p is a prime of the form 4q + 3 . Run the program for n = 5, 13, 17, 29, and 37 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 1 . Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n?

Go to https://www.d.umn.edu/~jgallian/compsciProject2018/html/chap12/ch12ex6.html