# Chapter 13 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 = 2 k . 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 = p k 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 install java SDK or JDK on your computer if you could not see the program below.)

Example: if you want to run this software for n = 4, 8, 16, and 32, please input the number individually into the text box above and press the "Enter" button.