Transcript Proofs, Recursion and Analysis of Algorithms

Sets, Combinatorics, Probability, and
Number Theory
Mathematical Structures
for Computer Science
Chapter 3
Copyright © 2006 W.H. Freeman & Co.
MSCS Slides
Probability
Fundamental Theorem of Arithmetic

FUNDAMENTAL THEOREM OF ARITHMETIC
For every integer n  2, n is a prime number or can be
written uniquely (ignoring ordering) as a product of
prime numbers.
We ignore the order in which we write the factors:
2(3)(3) = 3(2)(3)

If a and b are positive integers, then gcd(a,b) can
always be written as a linear combination of a and b,
that is, gcd(a,b) = ia + jb for some integers i and j.

Section 3.7
gcd(420,66) = 6 = 3(420)  19(66)
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Fundamental Theorem of Arithmetic

The values 3 and 19 in gcd(420,66) = 3(420)  19(66)
are derived from the successive divisions done by the
Euclidean algorithm:
420 = 6 * 66 + 24
66 = 2 * 24 + 18
24 = 1 * 18 + 6
18 = 3 * 6 + 0

Rewriting the first three equations from the bottom up:
6 = 24 * 1 + 18
18 = 66 * 2 + 24
24 = 420 * 6 + 66

Now we use these equations in a series of substitutions:
6 = 24  1 * 18 = 24  1 * (66  2 * 24) (substituting for 18)
= 3 * 24  66
= 3 * (420  6 * 66)  66 (substituting for 24)
= 3 * 420  19 * 66
Section 3.7
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Fundamental Theorem of Arithmetic

THEOREM ON gcd(a, b)
Given positive integers a and b, gcd(a,b) is the linear
combination of a and b that has the smallest positive
value.
From the theorem on gcd(a,b), it follows that a and b
are relatively prime if and only if there exist integers i
and j such that:
ia + jb = 1
DEFINITION: RELATIVELY PRIME
Two integers a and b are relatively prime if gcd(a,b)
1.
Section 3.7
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Fundamental Theorem of Arithmetic

THEOREM ON DIVISION BY PRIME
NUMBERS
Let p be a prime number such that p  ab, where a and
b are integers. Then, either p  a or p  b.
To find the unique factorization of 825 as a product of
primes, we can start by simply dividing 825 by
successively larger primes:
825 = 3 * 275 = 3 * 5 * 55 = 3 * 5 * 5 * 11 = 3 * 52 * 11

Doing the same on 455:
455 = 5 * 7 * 13

Section 3.7
From these factorizations, we can see that gcd(825,
455) = 5.
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More on Prime Numbers

Section 3.7
THEOREM ON SIZE OF PRIME FACTORS
If n is a composite number, then it has a prime factor
less than or equal to (n)1/2.
Given n = 1021, let’s find the prime factors of n or
determine that n is prime. The value of (1021)1/2 is just
less than 32. So the primes we need to test are 2, 3, 5,
7, 11, 13, 17, 19, 23, 29, 31. None divides 1021, so
1021 is prime.
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More on Prime Numbers

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Section 3.7
THEOREM ON INFINITY OF PRIMES
(EUCLID)
There is an infinite number of prime numbers.
Assume that there is a finite number of primes. Let the
value of s = the sum of all primes + 1. Therefore, s is
not prime. Thus, s is composite and by the
fundamental theorem of arithmetic, s can be factored
as a product of (some of) the prime numbers.
Suppose that pj is one of the prime factors of s, that is,
s = pj (m) for some integer m. Then:
1 = s – p1 p2 … pk = pj(m)  p1 p2 … pk
= pj (m  p1 … pj  1 pj + 1 …pk)
Therefore, pj  1, which is a contradiction.
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Euler Phi Function

DEFINITION: EULER PHI FUNCTION
For n an integer, n  2, the Euler (pronounced
“oiler”) phi function of n, (n), is the number of
positive integers less than or equal to n and relatively
prime to n. ((n) is pronounced “fee” of n.)
For example:

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
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
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Section 3.7
(2) = 1 (the number 1)
(3) = 2 (the numbers 1, 2)
(4) = 2 (the numbers 1, 3)
(5) = 4 (the numbers 1, 2, 3, 4)
(6) = 2 (the numbers 1, 5)
(7) = 6 (the numbers 1, 2, 3, 4, 5, 6)
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