Exercise 6. This software determines the number of solutions to the equation $$X^2 = -I$$ where $$X$$ is a $$2 \times 2$$ 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 $$n = 2k$$ 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 $$n$$ ?

Please enter $$n$$ and click the button, the solutions will show below.