Definitions: Prime and Composite
Primes and Greatest Common Divisors
Discrete Mathematics I — MATH/COSC 1056E
Julien Dompierre
Department of Mathematics and Computer Science
Laurentian University
Sudbury, October 15, 2008
Eratosthenes
Definition
A positive integer p greater than 1 is called prime if the only
positive factors of p are 1 and p.
A positive integer greater than 1 which is not a prime is called
composite.
Note: An integer n is composite if and only if there exists an
integer a such that a | n and 1 < a < n.
The primes less than 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
The Sieve of Eratosthenes
Born: 276 BC in Cyrene, North
Africa (now Shahhat, Libya).
Died: 194 BC in Alexandria,
Egypt.
He was a Greek mathematician,
poet, athlete, geographer and
astronomer. He was the first
person to calculate the circumference of the Earth.
www-groups.dcs.st-and.ac.uk/
~history/Mathematicians/
Eratosthenes.html
The sieve of Eratosthenes is used to find the primes not exceeding
a specified positive integer.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Sieve of Eratosthenes — 2
The Sieve of Eratosthenes — 3
Integers divisible by 2, other than 2, receive an underline.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Sieve of Eratosthenes — 4
Integers divisible by 3, other than 3, receive an underline.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Sieve of Eratosthenes — 5
Integers divisible by 5, other than 5, receive an underline.
Integers divisible by 4, other than 4, receive an underline.
This step can be skipped because 4 already has an underline, then
4 is not prime number.
Also, all the multiples of 4 have already an underline.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Sieve of Eratosthenes — 6
The Sieve of Eratosthenes — 7
Integers divisible by 7, other than 7, receive an underline.
Integers divisible by 6, other than 6, receive an underline.
This step can be skipped because 6 already has an underline, then
6 is not prime number.
Also, all the multiples of 6 have already an underline.
The Sieve of Eratosthenes — 8, 9, 10, 11
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Sieve of Eratosthenes
Red integers are the primes ≤ 100.
Integers divisible by 8, 9 and 10, other than 8, 9 and 10, have
already an underline and then are not prime and their multiples
also have already an underline.
Integers divisible by 11, other than 11, receive
√ an underline. This
step is not mandatory because 11 > 10 = 100.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
Three Questions about Primes
1. In the sieve of Eratosthenes, why it is sufficient to stop at 10
to find all the primes not exceeding 100?
2. One can see that when integers get bigger, prime are more
sparse. Is there an integer big enough such that there is no
more prime past this number? Or is there an infinity of
primes?
3. What is the distribution of primes? How fast their occurrence
decreases?
Theorem
Theorem
If n is a composite integer, then n has a prime divisor less than or
√
equal to n.
Remark: The contrapositive of this theorem is: If n has no prime
√
divisor less than or equal to n, then n is not a composite integer.
The Fundamental Theorem of Arithmetic
Theorem (Fundamental Theorem of Arithmetic)
Every positive integer greater than 1 can be written uniquely as a
prime or as the product two or more primes where the prime
factors are written in order of non decreasing size.
The primes are the building blocks of positive integers, i.e., any
number is composed of primes.
The Infinitude of Primes
Theorem
There are infinitely many primes.
Euclid (c. −325 — −265) was the author of the most successful
mathematics book ever written. His Elements appeared in over
1000 different editions from ancient to modern times. The proof
by contradiction of this theorem come from Euclid’s book.
Distribution of Primes
Definition: The Greatest Common Divisor
Definition
Let a and b be integers, not both zero. The largest integer d such
that d | a and d | b is called the greatest common divisor of a
and b. The greatest common divisor of a and b is denoted by
gcd(a, b).
Theorem (The Prime Number Theorem)
The ratio
number of primes not exceeding x
x/ ln x
approaches 1 as x grows without bound.
Definition
This theorem was conjectured by Carl Friedrich Gauss and
Adrien-Marie Legendre but were not proved. Jacques Hadamard
and Charles-Jean-Gustave-Nicholas de la Vallée-Poussin proved it
in 1896.
The integers a and b are relatively prime if their greatest
common divisor is 1.
gcd and Prime Factorization
where each exponent is a non negative integer, and where all
primes occurring in the prime factorization of either a or b are
included in both factorizations, with zero exponents if necessary.
Then gcd(a, b) is given by
min(a1 ,b1 )
The integers a1 , a2 , . . . , an are pairwise relatively prime if
gcd(ai , aj ) = 1 whenever 1 ≤ i < j ≤ n.
Definition: Least Common Multiple
Another way to find the greatest common divisor of two integers is
to use the prime factorization if these integers. Suppose that the
prime factorizations of the integers a and b, neither equal to zero,
are
a = p1a1 p2a2 · · · pnan , b = p1b1 p2b2 · · · pnbn
gcd(a, b) = p1
Definition
min(a2 ,b2 )
p2
min(an ,bn )
· · · pn
.
Definition
The least common multiple of the positive integers a and b is
the smallest positive integer that is divisible by both a and b.
The least common multiple of a and b is denoted by lcm(a, b).
lcm and Prime Factorization
gcd and lcm
Another way to find the least common multiple of two integers is
to use the prime factorization if these integers. Suppose that the
prime factorizations of the integers a and b, neither equal to zero,
are
a = p1a1 p2a2 · · · pnan , b = p1b1 p2b2 · · · pnbn
where each exponent is a non negative integer, and where all
primes occurring in the prime factorization of either a or b are
included in both factorizations, with zero exponents if necessary.
Then lcm(a, b) is given by
max(a1 ,b1 )
lcm(a, b) = p1
max(a2 ,b2 )
p2
max(an ,bn )
· · · pn
.
Another method to find the lcm(a, b) is to find first the gcd(a, b)
and then to use the following theorem:
Theorem
Let a and b b positive integers. Then
ab = gcd(a, b) × lcm(a, b).