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.