Handout #30
CS103A
Feb. 13, 2008
Robert Plummer
Introduction to Number Theory
Basics of Number Theory
Number theory was once thought to be “pure” mathematics – math for the sake of math.
But with
the evolution of computers (and particularly cryptography), number theory has become known as
an applied area of math.
Much cryptology relies on some very old theorems from number theory.
What is number theory?
It is the branch of mathematics that studies problems about natural
numbers, integers and rational numbers, i.e., anything but real numbers.
It was invented by the
Greeks.
Number theory is a subject in which the concepts are simple but the problems can be
quite challenging.
We choose to work with number theory for two reasons in 103A: a) The simple
axioms and definitions in number theory allow us to focus on proof techniques and not get
distracted; b) Number theory has many important applications in CS.
We assume for our purposes that you understand what integers are: negative and positive whole
numbers including 0.
Note that 0 is considered to be neither positive nor negative.
We also
assume that you understand and accept as valid the basic arithmetic operations.
The slides from
today's lecture discuss these points in more detail.
The integers are
closed
under addition, subtraction and multiplication, meaning if we
add/subtract/multiply an integer and another integer, we get an integer.
The integers, however, are
not closed under division, which sets division apart from other operations.
We can still consider
“div” an integer operation if we define it in a special way.
(Note: where proofs are not given and
not covered in lecture, you are encouraged to find your own.)
Definition
: If a and b are integers and a
≠
0, then the statement that
a divides b
means that there is
an integer c such that b = ac.
Our notation for “a divides b” is a | b.
Theorem 30.1
: For all integers a, b, and d, if d | a and d | b then d | (a
±
b)
Theorem 30.2:
For all integers a, b, and c, if a | b then a | bc.
Proof:
Suppose a | b, then there exists an integer k such that ak = b.
Since (ak)c
= bc, we
can conclude that a | bc, since there exists an integer (kc) such that
a(kc) = bc.
Q.E.D.
Theorem 30.3:
For all integers a, b, and c, if a | b and b | c then a | c.