Chapter 4
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 <a,b>, <a>, <b>, and <a>
<b>. 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 |<a, b>|,
|<a>|,
|<b>|, and |<a> <b>|.
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 ?