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Prime and Composite Numbers
MATH 100 Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Prime and Composite Numbers
Number Theory
Number theory is a branch of mathematics devoted to the study
of properties of the natural numbers.
J. Robert Buchanan
Prime and Composite Numbers
Number Theory
Number theory is a branch of mathematics devoted to the study
of properties of the natural numbers.
Definition
The natural number a is divisible by the natural number b if
there exists a natural number k such that a = b k . If b divides a
we write b|a.
J. Robert Buchanan
Prime and Composite Numbers
Number Theory
Number theory is a branch of mathematics devoted to the study
of properties of the natural numbers.
Definition
The natural number a is divisible by the natural number b if
there exists a natural number k such that a = b k . If b divides a
we write b|a.
Definition
A natural number greater than 1 that has only itself and 1 as
divisors is called a prime number. A natural number greater
than 1 that is not prime is called composite.
J. Robert Buchanan
Prime and Composite Numbers
Sieve of Eratosthenes (1 of 2)
One method for determining if a natural number is prime.
1
List the natural numbers 2, 3, 4, . . . , N.
2
2 is prime, strike out every higher multiple of 2.
3
3 is prime, strike out every higher multiple of 3.
4
5 is prime, strike out every higher multiple of 5.
5
Keep√going until you reach the largest natural number less
than N.
6
All you have left is a list of prime numbers.
J. Robert Buchanan
Prime and Composite Numbers
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
25
35
45
J. Robert Buchanan
6
16
26
36
46
7
17
27
37
47
8
18
28
38
48
9
19
29
39
49
Prime and Composite Numbers
10
20
30
40
50
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
2
11
21
31
41
3
13
23
33
43
5
15
25
35
45
J. Robert Buchanan
7
17
27
37
47
9
19
29
39
49
Prime and Composite Numbers
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
2
11
31
41
3
13
23
5
25
35
43
J. Robert Buchanan
7
17
37
47
19
29
49
Prime and Composite Numbers
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
2
11
31
41
3
13
23
5
43
J. Robert Buchanan
7
17
37
47
19
29
49
Prime and Composite Numbers
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
2
11
31
41
3
13
23
5
43
J. Robert Buchanan
7
17
19
29
37
47
Prime and Composite Numbers
Sieve of Eratosthenes (2 of 2)
1
2
3
We would like to find all the prime numbers less than 50.
Start with a list of natural numbers 2, 3, . . . , 50.
√
Use the sieve procedure up to 50 ≈ 7.
2
11
31
41
51
61
71
81
91
52
62
72
82
92
5
3
13
23
43
53
63
73
83
93
17
54
64
74
84
94
55
65
75
85
95
56
66
76
86
96
37
47
57
67
77
87
97
19
29
58
68
78
88
98
59
69
79
89
99
60
70
80
90
100
Now find the prime numbers less than 100. Use your i>clicker2
to submit the count of prime numbers less than 100.
J. Robert Buchanan
Prime and Composite Numbers
Divisibility Tests
Divisible by
2
3
4
5
6
8
9
10
12
Test
Last digit is 0, 2, 4, 6, or 8.
Sum of the digits is divisible by 3.
Last two digits form a number divisible by 4.
Last digit is 0 or 5.
Number is divisible by 2 and 3.
Last three digits form a number divisible by 8.
Sum of the digits is divisible by 9.
Last digit is 0.
Number is divisible by 4 and 3.
J. Robert Buchanan
Prime and Composite Numbers
Example
Consider the number 45815. Determine if it is divisible by
n
2
3
4
5
6
8
9
10
12
(n | 45815)?
J. Robert Buchanan
Prime and Composite Numbers
Example
Consider the number 45815. Determine if it is divisible by
n
2
3
4
5
6
8
9
10
12
(n | 45815)?
No
No
No
No
No
No
J. Robert Buchanan
Prime and Composite Numbers
Example
Consider the number 45815. Determine if it is divisible by
n
2
3
4
5
6
8
9
10
12
(n | 45815)?
No
No
No
No
No
No
No
No
J. Robert Buchanan
Prime and Composite Numbers
Example
Consider the number 45815. Determine if it is divisible by
n
2
3
4
5
6
8
9
10
12
(n | 45815)?
No
No
No
Yes
No
No
No
No
No
J. Robert Buchanan
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Leap Years
2012
2000
2104
1532
2224
J. Robert Buchanan
Not Leap Years
2014
1800
2106
1500
2222
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Which if the following years is/was/will be a leap year? Use your
i>clicker2 to select “A” for leap year and “B” for not a leap year.
1780
J. Robert Buchanan
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Which if the following years is/was/will be a leap year? Use your
i>clicker2 to select “A” for leap year and “B” for not a leap year.
1780
1900
J. Robert Buchanan
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Which if the following years is/was/will be a leap year? Use your
i>clicker2 to select “A” for leap year and “B” for not a leap year.
1780
1900
2002
J. Robert Buchanan
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Which if the following years is/was/will be a leap year? Use your
i>clicker2 to select “A” for leap year and “B” for not a leap year.
1780
1900
2002
2100
J. Robert Buchanan
Prime and Composite Numbers
Leap Years
A leap year is a year which is divisible by 4 but not by 100
except if it is divisible by 400.
Which if the following years is/was/will be a leap year? Use your
i>clicker2 to select “A” for leap year and “B” for not a leap year.
1780
1900
2002
2100
2400
J. Robert Buchanan
Prime and Composite Numbers
Fundamental Theorem of Arithmetic
Every natural number can be expressed in one and only one
way as a product of primes (if the order of the factors is
disregarded). This unique product of primes is called the prime
factorization of the natural number.
J. Robert Buchanan
Prime and Composite Numbers
Fundamental Theorem of Arithmetic
Every natural number can be expressed in one and only one
way as a product of primes (if the order of the factors is
disregarded). This unique product of primes is called the prime
factorization of the natural number.
Example
Find the prime factorization of 885.
J. Robert Buchanan
Prime and Composite Numbers
Fundamental Theorem of Arithmetic
Every natural number can be expressed in one and only one
way as a product of primes (if the order of the factors is
disregarded). This unique product of primes is called the prime
factorization of the natural number.
Example
Find the prime factorization of 885.
885 = 3 · 5 · 59
J. Robert Buchanan
Prime and Composite Numbers
How Many Numbers Divide N?
To determine the number of divisors of N:
1
Write N as a product of prime factors using exponents.
2
Add 1 to each exponent.
3
Multiply these augmented exponents.
J. Robert Buchanan
Prime and Composite Numbers
How Many Numbers Divide N?
To determine the number of divisors of N:
1
Write N as a product of prime factors using exponents.
2
Add 1 to each exponent.
3
Multiply these augmented exponents.
Example
How many numbers divide 308?
J. Robert Buchanan
Prime and Composite Numbers
How Many Numbers Divide N?
To determine the number of divisors of N:
1
Write N as a product of prime factors using exponents.
2
Add 1 to each exponent.
3
Multiply these augmented exponents.
Example
How many numbers divide 308?
308 = 22 · 7 · 11 = 22 · 71 · 111
Number of divisors = (2 + 1)(1 + 1)(1 + 1) = (3)(2)(2) = 12
J. Robert Buchanan
Prime and Composite Numbers
How Many Numbers Divide 456?
456 = 23 · 31 · 191
J. Robert Buchanan
Prime and Composite Numbers
How Many Numbers Divide 456?
456 = 23 · 31 · 191
Number of divisors = (3 + 1)(1 + 1)(1 + 1) = (4)(2)(2) = 16
J. Robert Buchanan
Prime and Composite Numbers
Examples
Find the number of divisors of the following and submit that
number using your i>clicker2:
2520
J. Robert Buchanan
Prime and Composite Numbers
Examples
Find the number of divisors of the following and submit that
number using your i>clicker2:
2520
4050
J. Robert Buchanan
Prime and Composite Numbers
Infinitude of the Primes
Theorem
There are infinitely many prime numbers.
J. Robert Buchanan
Prime and Composite Numbers
Infinitude of the Primes
Theorem
There are infinitely many prime numbers.
Proof.
Suppose there were only finitely many primes,
p1 , p2 , . . . , pn .
Form the number N = (p1 · p2 · · · pn ) + 1
N > pi for i = 1, 2, . . . , n but pi does not divide N.
N is either prime or there is another prime (not in the
original list) which divides N.
J. Robert Buchanan
Prime and Composite Numbers
Searching for Prime Numbers
Large prime numbers are useful in cryptography.
Definition
For n = 1, 2, 3, . . ., the Mersenne numbers are those
generated by the formula
Mn = 2n − 1.
1
If n is composite, then Mn is also composite.
2
If n is prime, then Mn may be prime or composite.
The prime values of Mn are called the Mersenne primes.
J. Robert Buchanan
Prime and Composite Numbers
Some Mersenne Numbers
n
2
3
5
13
29
67
Mn
3
7
31
8,191
536,870,911
147,573,952,589,676,412,927
Prime?
Prime
Prime
Prime
Prime
Composite
Composite
In 2014 the largest known Mersenne prime number is
257885161 − 1 ≈ 5.818872662322464 × 1017425169
which has 17, 425, 170 digits. It was discovered January 25,
2013 (only the 48th Mersenne prime ever found).
J. Robert Buchanan
Prime and Composite Numbers
Fermat Numbers
Pierre de Fermat (1601–1665) conjectured that the formula
n
Fn = 22 + 1
would always produce a prime number. He checked
n = 0, 1, 2, 3, 4.
J. Robert Buchanan
Prime and Composite Numbers
Fermat Numbers
Pierre de Fermat (1601–1665) conjectured that the formula
n
Fn = 22 + 1
would always produce a prime number. He checked
n = 0, 1, 2, 3, 4.
n
0
1
2
3
4
5
Fn
3
5
17
257
65537
4294967297
J. Robert Buchanan
Prime?
Prime and Composite Numbers
Fermat Numbers
Pierre de Fermat (1601–1665) conjectured that the formula
n
Fn = 22 + 1
would always produce a prime number. He checked
n = 0, 1, 2, 3, 4.
n
0
1
2
3
4
5
Fn
3
5
17
257
65537
4294967297
J. Robert Buchanan
Prime?
Yes
Yes
Yes
Yes
Yes
Prime and Composite Numbers
Fermat Numbers
Pierre de Fermat (1601–1665) conjectured that the formula
n
Fn = 22 + 1
would always produce a prime number. He checked
n = 0, 1, 2, 3, 4.
n
0
1
2
3
4
5
Fn
3
5
17
257
65537
4294967297
J. Robert Buchanan
Prime?
Yes
Yes
Yes
Yes
Yes
(641)(6700417)
Prime and Composite Numbers
Euler’s Formula
Leonhard Euler noted in 1732 that the formula
p(n) = n2 − n + 41
always produces a prime number for 1 ≤ n < 41, but is
composite when n = 41.
n
1
5
9
13
..
.
p(n)
41
61
113
197
..
.
n
2
6
10
14
..
.
p(n)
43
71
131
223
..
.
n
3
7
11
15
..
.
p(n)
47
83
151
251
..
.
n
4
8
12
16
..
.
p(n)
53
97
173
281
..
.
37
1373
38
1447
39
1523
40
1601
J. Robert Buchanan
Prime and Composite Numbers
Escott’s Formula
E.B. Escott offered in 1879 the formula
q(n) = n2 − 79n + 1601
which always produces a prime number for 1 ≤ n < 80, but is
composite when n = 80.
n
1
5
9
13
..
.
q(n)
1523
1231
971
743
..
.
n
2
6
10
14
..
.
q(n)
1447
1163
911
691
..
.
n
3
7
11
15
..
.
q(n)
1373
1097
853
641
..
.
n
4
8
12
16
..
.
q(n)
1301
1033
797
593
..
.
76
1373
77
1447
78
1523
79
1601
J. Robert Buchanan
Prime and Composite Numbers