**Exercise 5.**
This software finds the nilpotent elements in \(Z_n[i] = \{ a + bi \;|\; a \) and \(b\) both belong to \(Z_n \}\). Run the software for \(n = 4, 8, 16,\) and \(32\). Make a conjecture about the number of nilpotent elements when \(n = 2k\). Run the software for \(n = 3, 5, 7, 11, 13,\) and \(17\). What do these values for \(n\) have in common? Make a conjecture about the number of nilpotent elements for these \(n\) . Run the program for \(n = 9\). Do you need to revise the conjecture you make based on \(n = 3, 5, 7, 11, 13\), and \(17\)? Run the software for \(n = 9, 25\), and \(49\). What do these values for \(n\) have in common? Make a conjecture about the number of nilpotent elements for these \(n\) . Run the program for \(n = 27\). Do you need to revise the conjecture you made based on \(n = 9, 25,\) and \(49\)? Run your program for \(n = 125\) (this may take a few seconds.) On the basis of all of your data for this exercise make a single conjecture in the case that \(n = pk\) where \(p\) is any prime. Run the program for \(n = 6, 15,\) and \(21\). Make a conjecture. Run the program for \(12, 20, 28\), and \(45\). Make a conjecture. Run the program for \(36\) and \(100\) (this may take a few minutes). On the basis of all your data for this exercise make a single conjecture that covers all integers \(n > 1\).

Please enter \(n\) and click the button, the result will show below.