Exercise 2. Let Zn[i] = { a+bi | a, b belong to Zn, i2=-1 } (the Gaussian integers modulo n ). This software finds the group of units of this ring and the order of each element of the group. Run the program for n = 3, 7, 11, and 23. Is the group of units cyclic for these cases? Try to guess a formula for the order of the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 3. Run the program for n = 9 and 27. Are the groups cyclic? Try to guess a formula for the order when n = 3k. Run the program for n = 5, 13, 17, and 29. Is the group cyclic for these cases? What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 1. Try to guess a formula for the largest order of any element in the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 1. On the basis of the orders of the elements of the group of units, try to guess the isomorphism class of the group. Run the program for n = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the isomorphism class of the group.

Please input the set size n here (n < 30):