Theory of Computation
Bezout's Identity, Euclid's 1st Theorem, Unique Factorization Theorem, Pairing Functions
Vladimir Kulyukin
www.vkedco.blogspot.comwww.vkedco.blogspot.com
Outline● Number Theory Concepts & Theorems
Bezout's Identity Euclid's 1st Theorem Fundamental Theorem of Arithmetic (Unique Factoriza-
tion Theorem)● Pairing Functions
Number Theory Concepts &
Theorems
Bezout’s Identity
.
such that and integers are then there
,gcd i.e., , isdivisor common
greatest whoseintegers are and If
dbyax
yx
da,bd
ba
Bezout’s Identity: Example
642112)3(
1,3
6)1(42412
1,4
64212
6)42,12gcd(
yx
yx
yx
Euclid’s 1st Theorem (Book VII of Euclid’s Elements)
If a prime divides the product of two integers, then the prime divides at least one of the two integers. Formally, if p|ab, then p|a or p|b, where p is a prime and a and b are integers.
Proof Technique Note 1
● Suppose we want to prove a statement: if A then B or C
● We assume A and not B and prove C ● In other words, if A and not B are true, then C must be
true, because otherwise, B or C cannot be true● We can also assume A and not C and prove B
Euclid’s 1st Theorem: Proof
. offactor a is that
shown and )|( that assumed also have could We
. offactor a is So
).()(
Now, .1such that and integers are there
Identity, sBezout'By .1gcdThen .number somefor
, ,| Since ).|( prime, is ,| that Assume
ap
bp
bp
rybxprpybpxbaybpxaypxbb
aypxyx
(p,a)r
abrpabpappabp
Fundamental Theorem of Arithmetic
Every positive integer greater than 1 is either a prime or can be written as a product of primes. The factorization is unique except for the order of factor primes.
This theorem is also known as Unique Factorization Theorem
FTA: Examples
24
2
1111
3
5321200
3212
255210
22228
326
FTA: Key Insight
].2,0[],1,0[],4,0[
where,532 form theof is 1200 ofdivisor
any that knowy immediatel Then we .5321200 that know
weexample,For number. that ofdivisor every of formula the
know wenumber, a ofion factorizat prime theknow weIf
24
zyx
zyx
FTA: Proof Sketch
We need to prove 2 statements:1. Every natural number greater than 1 has a prime
factorization, i.e., can be written as a product of primes
2. The prime factorization is unique
FTA: Proof (Part 1)
. and of ionsfactorizat
prime theof consistsion that factorizat prime a has But then 7.
ions.factorizat prime haveboth and
ion,factorizat prime a havenot doesat number thsmallest
theis and than less numbers positive are and Since 6.
.1 and 1 where, composite, a isn Since 5.
composite. a is So 4.
ion.factorizat its as itself have it would it were, if because, prime, anot is 3.
.number thisCall
number.such smallest thebemust thereprinciple, ordering- wellBy the 2.
ion.factorizat prime a has 1an greater thnumber naturalevery not Suppose 1.
ba
n
ba
nn ba
nb na abn
n
n
n
Proof Technique Note 2
● Suppose that we want to prove that some mathematical object A is unique
● A common way of doing this is to postulate the existence of another mathematical object B with A’s properties and then show that A and B are the same
● In other words, A is unique, because any other object that has the same properties is A itself
FTA: Proof (Part 2)
.,,1for and that case thebemust it , since But, 7.
...., ,,for repeated becan trick same The 6.
.by both themdividingby
and ofout taken becan Thus, .2or that case the
bemust it prime, a is Since .or Theorem,1st sEuclid'By 5.
Thus, . that know We. Take 4.
3.
. and
ionsfactorizat prime twohas that 1an greater thnumber natural a be Let 2.
.or then , and prime a is if :Theorem1st sEuclid' Recall .1
21
32
12
11111
12111
1111
1211
21
ji
n
i
m
m
mn
qpnjimnFF
ppp
pF
Fpmi , qpqp
p...q|qp|qp
....q|qp|npp
n....qqF...ppFn
FF
n
p|bp|ap|abp
Pairing Functions
Pairing Functions
● Pairing functions are coding devices for mapping pairs of natural numbers into single natural numbers and vice versa
● Once we have pairing functions we will be able to map lists of numbers into single numbers and vice versa
● Mapping lists of numbers into a single number will give us a means to compile L programs into natural numbers
Pairing Functions
1221,
01122
1122,
yyx
y
yyx
x
x
x
Equation 8.1 (Ch. 3)
., tofor solution
unique a is therenumber, natural a a is If
zyxx, y
z
Equation 8.1 (Ch. 3)
2
12
1
2
112 .5
.1|2such that number largest theis s,other wordIn
.1z| 2max .4
).12(21z .3
).12(21, .2
, .1
x
x
x
d
d
x
x
z
yz
y
zx
x
y
yyx
zyx
Equation 8.1 (Ch. 3): Upper Bound for x & y
., Hence,
.1,1 Therefore,
.12211,
,1122, Since
zyzx
zyzx
yzyx
yzyxx
x
Example 1
.10111115225,0 :Check
.51112
011|2max
11122
10112210,
.10, Solve
0
yy
xx
y
yyx
yx
d
d
x
x
Example 2
.19120112222,2:Check
.251220122
220|2max
20122
191122,
.19, Solve
2
2
yyy
x
y
yyx
yx
d
d
x
x
Splitting Natural Numbers into
Left & Right Constituents
Equation 8.1 (Ch. 03)
.22,219;22,219
.55,010;05,010
:Examples
.,
.,
. , and :functions twodefines 8.1Equation
rrll
rrll
yyxrzr
xyxlzl
Nzzrzl
Lemma
recursive. primitive are and zrzl
Proof
. ,min)(
; ,min)(
Thus, .,,, then , If
yxzxzr
yxzyzl
zyzxyxzNz
zzy
zzx
Theorem 8.1 (Ch. 03)
zzrzzl
zzrzl
yyxrxyxl
zrzlx,y
, 4.
, 3.
,,, 2.
recursive primitive areThey 1.
:properties following thehave ,, functions The
Proof 8.1 (Ch. 03)
This theorem summarizes the properties of the pairing function and the splitting functions l(z) and r(z). Properties 2, 3, 4 follow from Equation 8.1. Property 1 follows from Equation 8.1 and the definitions for l(z) and r(z).
Reading Suggestions
● Ch. 03, Computability, Complexity, and Languages, 2nd Edition, by Davis, Sigal, Weyuker, Academic Press
● Great wiki article on Bezout's identity
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