Solved: Exercise 1. This software determines if U(n) is

Chapter 4, Problem 2CE

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Problem 2CE

Exercise 1. This software determines if U(n) is cyclic. Run the program for n = 8, 16, 32, 64, and 128. Make a conjecture. Run the program for 3, 9, 27, 81, 343, 5, 25, 125, 7, 49, 11, and 121. Make a conjecture. Run the program for n = 12, 20, 28, 44, 52, 15, 21, 33, 39, 51, 57, 69, 35, 55, 65, and 85. Make a conjecture.

Exercise 2. For any pair of positive integers m and n, let Zm + Zn= {(a,b) | a Zm, b Zn}. For any pair of elements (a,b) and (c,d) in Zm + Zn, define (a,b) + (c,d) = ((a+c) mod m, (b + d) mod n). [For example, in Z3 + Z4, we have (1, 2) + (2, 3) = (0, 1).] This software checks whether or not Zm + Zn is cyclic. Run the program for the following choices for m and n: (2, 2), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8), (3, 9), and (4, 6). On the basis of this output, guess how m and n must be related for Zm + Zn to be cyclic.

Exercise 3. In this exercise we assume a, b U(n). Define <a,b> = {ai bi | 0 <= i < |a|, 0 <= j < |b|}. This software computes the orders of , , , and . Run the program for the following choices for a, b, and n: (21, 101, 550), (21, 49, 550), (7, 11, 100), (21, 31, 100), and (63, 77, 100). On the basis of the output, make a conjecture about arithmetic relationships among ||, ||, ||, and | |.

Exercise 4. For each positive integer n, this software gives the order of U(n) and the order of each element in U(n). Do you see any relationship between the order of U(n) and the order of its elements? Run the program for n = 8, 16, 32, 64 and 128. Make a conjecture about the number of elements of order 2 in U(2k) when k is at least 3. Make a conjecture about the number of elements of order 4 in U(2k) when k is at least 4. Make a conjecture about the number of elements of order 8 in U(2k) when k is at least 5. Make a conjecture about the maximum order of any element in U(2k) when k is at least 3. Try to find a formula for an element of order 4 in U(2k) when k is at least 4.

Exercise 5. For each positive integer n, this software lists the number of elements of U(n) of each order. For each order d of some element of U(n), this software lists phi(d) and the number of elements of order d. (Recall that phi(d) is the number of positive integers less than or equal to d and relatively prime to d). Do you see any relationship between the number of elements of order d and phi(d)? Run the program for n = 3, 9 , 27, 81, 5, 25, 125, 7, 49, and 243. Make a conjecture about the number of elements of order d and phi(d) when n is a power of an odd prime. Run the program for n = 6, 18, 54, 162, 10, 50, 250, 14, 98, and 686. Make a conjecture about the number of elements of order d and phi(d) when n is twice a power of an odd prime. Make a conjecture about the number of elements of various orders in U(pk) and U(2pk) where p is an odd prime.

Exercise 6. For each positive integer n, this software gives the order of U(n). Run the program for n = 9, 27, 81, and 243. Try to guess a formula for the order of U(3k) when k is at least 2. Run the program for n = 18, 54, 162, and 486. How does the order of U(2x3k) appear to be related to the order of U(3k)? Run the program for n = 25, 125, and 625. Try to guess a formula for the order of U(5k) when k is at least 2. Run the program for n = 50, 250, and 1250. How does the order of U(2x5k) appear to be related to the order of U(5k)? Run the program for n = 49 and 343. Try to guess a formula for the order of U(7k) when k is at least 2. Run the program for n = 98 and 686. How does the order of U(2x7k) appear to be related to the order of U(7k)? Based on your guesses for U(3k),U(5k) and U(7k) guess a formula for the order of U(pk) when p is an odd prime and k is at least 2. What about the order of U(2xpk) when p is an odd prime and k is at least 2. Does your formula also work when k is 1 ?

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