1
Chapter
11
:
Euler’s
Phi Function
Practice HW
p.
80
#
1
,
2
a
, 3, 5 (Additional Web Exercises)
In
this chapter, we want to look at how to compute efficiently the number
, which
, is the number of integers betwe
en 1 and
m
that are relatively prime to
m
. That is,
We would like to have a method of computing
when
m
is larger. The next theorems
describe some efficient ways of doing this.
Theorem
1
:
If
p
is a prime n
umber, then
.
Proof:
█
Example 1
:
Compute
and
.
Solution:
█
2
Theorem 2
:
If
p
is a prime number, then
.
Proof:
The only divisors of
are 1 and the powers of
p
less than
k
, that is
for
. Let
a
be an integer where
. If
, then
must have a
factor of
, that is
. Thus, a number
a
is not relatively prime (
when
. Th
us
If
, then
for some integer
k
. The multiples of
p
between 1 and
are
,
which is
multiples of
p
. Hence,
and thus
.
█
Example 2
:
Compute
.
Solution:
█
We next state
and prove a lemma that will be useful in a later result.
3
Lemma:
For integers
a, m,
and
n,
if and only if
and
.
Proof:
Assume
and suppose
. Then
and
.
Hence
. This contradicts the fa
ct that
. A similar argument can be
made if we assume
.
Thus
and
Now assume
and
and suppo
se
. Then
and
. By the Fundamental Theorem of Arithmetic,
d
must have a prime divisor
p
where
. Thus,
and
. By the p
rime divisibility property, that says that
or
. If
, since
this contradicts the fact that
. Similarly, if
, since
this contradicts the fact that
.
In either case, we have a
contradiction and hence
.
█
For example, if
a
= 5,
m =
6, and
c
= 7, the lemma implies
W
e next use the previous lemma to prove a fundamental result.
Theorem 3
:
For two positive integers
m
and
n
, if the
, then
.
Proof:
We rearrange the integers between 1 and
mn
into an
array with
n
rows and
m
columns.
By Lemma 5,
Consider the
column of the array. Each entry in the
column when paired with
m
,
has the same greatest common
divisor.
Continued on Next Page
4
That is, if
,
(This can be seen by
calculating both using the first step of the Euclidean Algorithm). Since all elements in
each column have the same greatest common
divisor,
of the columns will have the
elements in the array that are relatively prime to
m
.
Now, consider the
columns. We must now show that each of these columns has
elements relat
ively prime to
n
, thus giving a total of
total elements
relatively prime to both
m
and
n
and hence
by the previous lemma. Consider the
column
Claim: In modu
lo
n
arithmetic, all entries in this column are just a rearrangement of
, that is, each entry in the column is in a the same distinct congruence class
as one of the integers
. For if not, there would exist inte
gers
such that
This implies that
Since by the Theorem assumption,
, this implies that
which is
impossible since
. Hence, since the
column elements
and
are the same elements with respect to
congruence in modulo
n
arithmetic and there are
elements relatively prime to
n
i
n list
, there are
elements in the
column
relatively prime to
n
. Thus there are
columns in the array containing the elements
relatively p
rime to
n
, and in each of these
columns
entries relatively prime to
n
.
This gives a total of
total elements that are relatively prime to both
m
and
n.
Since, by the previous lemma, the
se are the same elements relatively prime to
, we have
the result
█
5
Example 3
:
Compute
.
Solution:
█
Theorem 4
:
If
m
has the
prime factorization
, then
.
Proof:
We can prove this result using mathematical induction. For the trivial case, that is,
if
, then using Theorem 13.6 we have
.
Now, assume the result is true if
m
is a product of
r
primes. We want to show the result is
true if
m
is a product of
r
+ 1 primes. Suppose
. Noting that
we have
Hence, by the princi
ple of mathematical induction, the result holds.
█
Corollary 1
:
If
p
and
q
are primes where
, then
.
Proof:
█
6
Example 4
:
Compute
.
Solution:
█
Example
5
:
Compute
.
Solution:
█
Example 6
:
Compute
.
Solution:
█
7
Euler Phi Function w
ith Maple
Note that the
numtheory
package must be loaded to the home directory using the
with
statement before the
phi
command can be used.
> with(numtheory):
Compute
.
> phi(35);
Compute
.
> phi(360);
Compute
.
> phi(1575);
8
Chinese Remainder Theorem
Theorem 5:
Chinese Remainder Theorem.
The system of linear congruences
*
where
(Moduli are pairwise relatively prime
) can be solved for an integer
x
modulus
. Moreover, if
y
is another solution to these congruences, then
.
Proof:
Let
for
.
Then
(Proof Exercise).
By the Linear Congruence Theorem, there exists
an integer
where
(Note
is the multiplicative inverse of
mod
Al
so,
whenever
. (Proof Exercise)
Let
Claim that
x
satisfies every linear congruence. To show, that the
j
th
arbitrary congruence
with modulus
. Then
Continued on Next Page
9
To show there are no other incongruent solutions, suppose
y
is another solution to *. Then
for all
,
. Since
, it follows that
Thus,
. Since
when
, then it follows that
Hence,
and
. This complete
s the proof.
█
Chinese Remainder Theorem Summary
To solve t
he system of linear congruences
We compute
where
each
comes from the right had side of the given c
ongruences
,
for
,
is the multiplicative inverse of
mod
, that is,
solves the congruence
for
.
10
Example 7:
Use the Chinese Remainder Theorem to solve
the system of congruences
,
,
Solution:
11
█
12
13
Using the Chinese Remainder Theorem in Maple
To solve
,
,
> chrem( [2, 4, 6], [3, 5, 19] );
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