Introduction to Number Theory I - Boise State University · 2011-08-29 · Introduction to Number...
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Outline Divisibility Greatest common divisor Euclidean algorithm Extended Euclidean algorithm Introduction to Number Theory I Introduction to Number Theory I
Transcript of Introduction to Number Theory I - Boise State University · 2011-08-29 · Introduction to Number...
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Introduction to Number Theory I
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Divisibility
Greatest common divisor
Euclidean algorithm
Extended Euclidean algorithm
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionLet a and b be integers. We say that a 6= 0 divides b if there is aninteger k such that a · k = b. The notation for this is a|b.
Example
7|63 because 7 · 9 = 63.
Note that every number divides zero since a · 0 = 0 for everyinteger a.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaFor integers a, b, c, the following hold:
I If a|b, then a|b · c for all c.
I If a|b and b|c, then a|c.
I If a|b and a|c, then a|(sb + tc) for all s and t.
I If a|b and b|a, then a = ±b.
I For all c 6= 0, a|b if and only if ca|cb.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaFor integers a, b, c, the following hold:
I If a|b, then a|b · c for all c.
I If a|b and b|c, then a|c.
I If a|b and a|c, then a|(sb + tc) for all s and t.
I If a|b and b|a, then a = ±b.
I For all c 6= 0, a|b if and only if ca|cb.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaFor integers a, b, c, the following hold:
I If a|b, then a|b · c for all c.
I If a|b and b|c, then a|c.
I If a|b and a|c, then a|(sb + tc) for all s and t.
I If a|b and b|a, then a = ±b.
I For all c 6= 0, a|b if and only if ca|cb.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaFor integers a, b, c, the following hold:
I If a|b, then a|b · c for all c.
I If a|b and b|c, then a|c.
I If a|b and a|c, then a|(sb + tc) for all s and t.
I If a|b and b|a, then a = ±b.
I For all c 6= 0, a|b if and only if ca|cb.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaFor integers a, b, c, the following hold:
I If a|b, then a|b · c for all c.
I If a|b and b|c, then a|c.
I If a|b and a|c, then a|(sb + tc) for all s and t.
I If a|b and b|a, then a = ±b.
I For all c 6= 0, a|b if and only if ca|cb.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionA number p > 1 with no positive divisors other than itself and 1 iscalled a prime number.
Non-prime numbers bigger than 1 are called composite numbers.
The Prime Number theorem:
For each positive integer n there are approximately n/ ln n primenumbers up to n.
Euclid’s theorem:
If p is a prime number and p divides a · b, then p divides a or pdivides b.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionA number p > 1 with no positive divisors other than itself and 1 iscalled a prime number.
Non-prime numbers bigger than 1 are called composite numbers.
The Prime Number theorem:
For each positive integer n there are approximately n/ ln n primenumbers up to n.
Euclid’s theorem:
If p is a prime number and p divides a · b, then p divides a or pdivides b.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
The Fundamental Theorem of Arithmetic:
Every positive integer n can be written in a unique way in the form
n = pe11 · pe2
2 · pe33 ....pek
k
where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
The Fundamental Theorem of Arithmetic:
Every positive integer n can be written in a unique way in the form
n = pe11 · pe2
2 · pe33 ....pek
k
where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
The Fundamental Theorem of Arithmetic:
Every positive integer n can be written in a unique way in the form
n = pe11 · pe2
2 · pe33 ....pek
k
where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).
DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.
Dirichlet’s theorem:
Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.
Example
gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).
DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.
Dirichlet’s theorem:
Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.
Example
gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).
DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.
Dirichlet’s theorem:
Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.
Example
gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaIf a = q · b + r then gcd(a, b) = gcd(b, r).
Theorem (Euclid, 300 B.C.)
Given integers a and b with b > 0, there exist unique integers qand r satisfying
a = q · b + r , 0 ≤ r < b
The integers q and r are called, respectively, the quotient andremainder in the division of a and b.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaIf a = q · b + r then gcd(a, b) = gcd(b, r).
Theorem (Euclid, 300 B.C.)
Given integers a and b with b > 0, there exist unique integers qand r satisfying
a = q · b + r , 0 ≤ r < b
The integers q and r are called, respectively, the quotient andremainder in the division of a and b.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example
gcd(1147, 899) =
gcd(899, 248︸︷︷︸1147=1·899+248
)
= gcd(248, 155︸︷︷︸899=3·248+155
)
= gcd(155, 93︸︷︷︸248=1·155+93
)
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example
gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248
)
=
gcd(248, 155︸︷︷︸899=3·248+155
)
= gcd(155, 93︸︷︷︸248=1·155+93
)
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example
gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248
)
= gcd(248, 155︸︷︷︸899=3·248+155
)
=
gcd(155, 93︸︷︷︸248=1·155+93
)
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example
gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248
)
= gcd(248, 155︸︷︷︸899=3·248+155
)
= gcd(155, 93︸︷︷︸248=1·155+93
)
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example (Cont.)
= gcd(93, 62︸︷︷︸155=1·93+62
)
=
gcd(62, 31︸︷︷︸93=1·62+31
)
= gcd(31, 0︸︷︷︸62=2·31+0
)
= gcd(31, 0)
= 31.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example (Cont.)
= gcd(93, 62︸︷︷︸155=1·93+62
)
= gcd(62, 31︸︷︷︸93=1·62+31
)
=
gcd(31, 0︸︷︷︸62=2·31+0
)
= gcd(31, 0)
= 31.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example (Cont.)
= gcd(93, 62︸︷︷︸155=1·93+62
)
= gcd(62, 31︸︷︷︸93=1·62+31
)
= gcd(31, 0︸︷︷︸62=2·31+0
)
=
gcd(31, 0)
= 31.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example (Cont.)
= gcd(93, 62︸︷︷︸155=1·93+62
)
= gcd(62, 31︸︷︷︸93=1·62+31
)
= gcd(31, 0︸︷︷︸62=2·31+0
)
= gcd(31, 0)
= 31.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.
1. Let r0 = a and r1 = b.
2. Set i=1.
3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,
ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri
4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.
5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.
The division step is executed in at most 2log2(b) + 1 times.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.
1. Let r0 = a and r1 = b.
2. Set i=1.
3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,
ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri
4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.
5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.
The division step is executed in at most 2log2(b) + 1 times.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.
1. Let r0 = a and r1 = b.
2. Set i=1.
3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,
ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri
4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.
5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.
The division step is executed in at most 2log2(b) + 1 times.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
LemmaThe following statements about the greatest common divisor hold:
I Every common divisor of a and b divides gcd(a, b).
I gcd(k · a, k · b) = k · gcd(a, b) for all k > 0.
I If gcd(a, b) = 1 and gcd(a, c) = 1,then gcd(a, b · c) = 1.
I If a|bc and gcd(a, b) = 1, then a|c.
Proof: Exercise.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
TheoremGiven integers a and b not both of which are zero, there existintegers x and y such that gcd(a, b) = a · x + b · y.
TheoremLet a and b be integers with at least one of them different fromzero. The integers a and b are relatively prime if and only if1 = a · x + b · y for some x , y ∈ Z.
Introduction to Number Theory I
OutlineDivisibility
Greatest common divisorEuclidean algorithm
Extended Euclidean algorithm
Example
gcd(12378, 3054) = 4 = 3186 · 12378 + (−12913) · 3054
TheoremLet a and b be positive integers. Then the equationax + by = gcd(a, b) always has a solution in integers x and y. If(x0, y0) is one solution, then every solution has the formx = x0 + b·k
gcd(a,b) and y = y0 − a·kgcd(a,b) for some k ∈ Z.
Proof: Exercise.
Introduction to Number Theory I
