This preview shows pages 1–10. Sign up to view the full content.
Chapter 3: Elementary Number Theory And
Methods of Proof
January 27, 2008
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Outline
1
3.1 Direct Proof and Counterexample I: Introduction
2
3.3 Direct Proof and Counterexample III: Divisibility
3
3.4 Direct Proof and Counterexample IV: Division into Cases
and the QuotientRemainder Theorem
4
3.5 Direct Proof and Counterexample V: Floor and Ceiling
5
3.6 Indirect Argument: Contradiction and Contraposition
6
3.8 The Euclidean Algorithm
•
In this Chapter, we will investigate some properties of the
set of integers
Z
and the set of rational numbers (quotients
of integers)
Q
•
At the same time, we will try to apply some of the methods
for proving statements which we have learned in Chapters
1 and 2.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document •
We will start by giving some basic deﬁnitions of number
theory which will be used repeatedly in the rest of this
chapter.
Deﬁnition
An integer is
even
if, and only if,
n
is two times some integer.
An integer
n
is
odd
if, and only if, it is two times some integer
plus 1.
n
is even
⇔ ∃
an integer
k
such that
n
=
2
k
n
is odd
⇔ ∃
an integer
k
such that
n
=
2
k
+
1
Examples
(a)
34 is an even integer
34
=
2
·
17
(b)
157 is an odd integer

157
=
2
·
(

79
) +
1
(c)
0 is an even integer
0
=
2
·
0
(d)
For any integers
m
,
n
, the integer 4
m
3
n
2
is even, since
4
m
3
n
2
=
2
(
2
m
3
n
2
)
(e)
For any two integers
m
and
n
, the integer 6
m
+
4
n
2
+
5 is
odd, since
6
m
+
4
n
2
+
5
=
2
(
3
m
+
2
n
2
+
2
) +
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Deﬁnition
An integer
n
is
prime
if, and only if,
n
>
1 and for all positive
integers
k
and
m
, if
n
=
k
·
m
then
k
=
1 or
m
=
1. An integer
n
is
composite
if, and only if,
n
>
1 and,
n
=
k
·
m
for some
k
6
=
1 and
m
6
=
1.
n
is prime
⇔ ∀
positive integers
k
,
m
, if
n
=
k
·
m
→
k
=
1 or
m
=
1
n
is composite
⇔ ∃
positive integers
k
,
m
such that
n
=
k
·
m
and
k
6
=
1
,
m
6
=
1
Proving Existential Statements
•
Suppose we are given a statement of the form
∃
x
∈
D
such that
P
(
x
)
•
Remember that this statement is true if, and only if,
P
(
x
)
is true for at least one
x
∈
D
•
So, in order to prove this statement, we need to ﬁnd one
such
x
∈
D
which will make
P
(
x
)
true.
•
Another way to prove such a statement is to give a method
(
algorithm
) for ﬁnding such an
x
.
•
Both these methods for proving the truth of an existential
statement are called
constructive proofs of existence
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Examples
(a)
Prove the following:
∃
an even integer
n
that can be written
in two ways as a sum of two prime numbers.
(b)
Suppose
m
and
n
are two integers. Prove the following:
∃
an integer
k
such that 14
m
+
26
n
2
=
2
k
Solution:
(a)
Let
n
=
10, then
10
=
3
+
7
10
=
5
+
5
(b)
Since 14
m
+
26
n
2
=
2
(
7
m
+
13
n
2
)
, we can take
k
=
7
m
+
13
n
2
•
Proofs of existential statements can also be
nonconstructive
. In that case, we show only that an
x
satisfying
P
(
x
)
must exist without actually exhibiting it or
giving a “recipe” (algorithm) as to how to ﬁnd it.
•
Also, a nonconstructive proof can also be done by
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/17/2008 for the course MTH 314 taught by Professor Forgot during the Spring '08 term at Ryerson.
 Spring '08
 forgot
 Number Theory, Division

Click to edit the document details