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.