CS103A
HO# 32
Introduction to Number Theory
2/11/08
1
CS103A
2/11/08
Problem Set 7 will be distributed Wednesday, 2/13
and it will be due Friday, 2/22.
Outline of topics for CS103A
Basic Tools
Formal Logic and Proof Techniques
Number Theory and its Applications
Proving “Real” Theorems
Induction
Program Proofs
Recursion
Combinatorics & Probability
Functions
Number Theory
We will not try to develop number theory from the
ground up, but we should acknowledge that this can
be done from a small number of axioms.
One example: Peano's Axioms, proposed by
Giuseppe Peano (1858 – 1932) in 1889.
Peano's Axioms
• There is a number 0.
• Every number has a successor, denoted by S(a).
• There is no number whose successor is 0, i.e.,
∀
x (S(x)
≠
0).
• Two numbers with the same successor are themselves
equal, i.e.,
∀
x
∀
y(S(x) = S(y)
→
x = y)
• If a property is possessed by 0 and if the successor of
every number possessing the property also possesses it,
then it is possessed by every number, i.e.,
[Q(0)
∧ ∀
x(Q(x)
→
Q(S(x))]
→ ∀
xQ(x)
What we assume
. . . 3
2
1
0
1
2
3 . . .
• The existence of
Z
, the set of all integers.
Z
= {… 3, 2, 1, 0, 1, 2, 3, …}
• Commutativity:
a + b = b + a
a
×
b = b
×
a
• Associativity:
(a + b) + c = a + (b + c)
(a
×
b) x c = a
×
(b
×
c)
• Distributivity: a
×
(b + c) = a
×
b + a
×
c
(a + b)
×
c = a
×
c + b
×
c
What we assume
• a + 0 = a
• a
×
1 = a
• Negative numbers: the equation a + x = 0 has
the unique solution x = a.
• Subtraction: x = b  a
⇔
a + x = b
• The integers are closed under addition,
multiplication, and subtraction, i.e., if a and b
are integers, then so are a + b, a
×
b, and a – b.
• The usual algebraic manipulations such as
factoring polynomials, raising to powers, etc.
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CS103A
HO# 32
Introduction to Number Theory
2/11/08
2
Division
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
.
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 Winter '07
 Plummer,R
 Computer Science, Number Theory, Natural number, Prime number, Euclid, gcd

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