Download investigation_prime_numbers

© T Madas
A prime number or simply a prime, is a
number with exactly two factors.
These two factors are always the number 1 and the
prime number itself
All prime numbers are odd except number 2
1 is not a prime number.
2 is the smallest prime
There is no largest prime.
There are infinite prime numbers
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
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
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
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
Cross off the number 1
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
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
Cross off all the multiples of 2 except 2
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
2
11
21
31
41
51
61
71
81
91
3
13
23
33
43
53
63
73
83
93
5
15
25
35
45
55
65
75
85
95
7
17
27
37
47
57
67
77
87
97
9
19
29
39
49
59
69
79
89
99
Cross off all the multiples of 3 except 3
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
2
11
31
41
61
71
91
3
13
23
43
53
73
83
5
25
35
55
65
85
95
7
17
37
47
67
77
19
29
49
59
79
89
97
Cross off all the multiples of 5 except 5
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
2
11
31
41
61
71
91
3
13
23
43
53
73
83
5
7
17
37
47
67
77
19
29
49
59
79
89
97
Cross off all the multiples of 7 except 7
© T Madas
The Sieve of Eratosthenes can be used to find the
prime numbers up to 100 very quickly
2
11
31
41
61
71
3
13
23
43
53
5
7
17
19
29
37
47
59
67
73
83
79
89
97
These are the prime numbers up to 100
© T Madas
The
Prime Numbers
up to 200
© T Madas
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Primes up to 200
101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190
191 192 193 194 195 196 197 198 199 200
© T Madas
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Primes up to 200
Cross off:
number 1
multiples of 2 except 2
multiples of 3 except 3
multiples of 5 except 5
multiples of 7 except 7
multiples of 11 except 11
multiples of 13 except 13
101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190
191 192 193 194 195 196 197 198 199 200
© T Madas
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Primes up to 200
Cross off:
number 1
multiples of 2 except 2
multiples of 3 except 3
multiples of 5 except 5
multiples of 7 except 7
multiples of 11 except 11
multiples of 13 except 13
101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190
191 192 193 194 195 196 197 198 199 200
© T Madas
Interesting Facts Involving Primes
© T Madas
Every even number other than 2, can be
written as the sum of two primes
16 = 3 + 13 = 5 + 11
22 = 3 + 19 = 11 + 11
40 = 3 + 37 = 11 + 29 = 17 + 23
52 = 5 + 47 = 11 + 41
Write these even numbers as the sum of two primes,
at least three different ways
50 = 3 + 47 = 7 + 43 = 13 + 37
100 = 3 + 97 = 11 + 89 = 17 + 83
150 = 11 + 139 = 13 + 137 = 19 + 131
200 = 3 + 197 = 7 + 193 = 13 + 187
© T Madas
Every even number other than 2, can be
written as the sum of two primes
This statement is known as the Goldbach conjecture.
In 1742 Christian Goldbach requested from Leonhard Euler,
the most prolific mathematician of all times, for a proof for his
conjecture.
Euler could not prove this statement, nor has anyone else to
this day, although no counter example can be found.
C Goldbach
1690 - 1764
L Euler
1707 - 1783
© T Madas
Every odd number other than 1, can be written
as the sum of a prime and a power of 2
3 = 2 + 20
17 = 13 + 22
35 = 31 + 22 = 19 + 24 = 3 + 25
81 = 79 + 21 = 17 + 26
Write these odd numbers as the sum of a prime and
a power of 2
25 = 23 + 21 = 17 + 23
75 = 73 + 21 = 71 + 22 = 67 + 23
125 = 109 + 24 = 64 + 26
175 = 173 + 21 = 167 + 23 = 47 + 27
© T Madas
Every even number can be written as the
difference of 2 consecutive primes
2 = 5–3 = 7–5
4 = 11 – 7 = 17 – 13
6 = 29 – 23 = 37 – 31 = 59 – 53
8 = 97 – 89
Write these even numbers as the difference of 2
consecutive primes
10 = 149 – 139
12 = 211 – 199
14 = 127 – 113
© T Madas
Every prime number greater than 3 is of the
form 6n ± 1, where n is a natural number
5 = 6x1–1
7 = 6x1+1
11 = 6 x 2 – 1
13 = 6 x 2 + 1
17 = 6 x 3 – 1
19 = 6 x 3 + 1
23 = 6 x 4 – 1
29 = 6 x 5 – 1
Careful because the converse
statement is not true:
Every number of the form 6n ± 1
is not a prime number
© T Madas
Every prime number of the form 4n + 1, where
n is a natural number, can be written as the
sum of 2 square numbers
5 = 4x1+1= 4+1
13 = 4 x 3 + 1 = 9 + 4
17 = 4 x 4 + 1 = 16 + 1
29 = 4 x 7 + 1 = 25 + 4
37 = 4 x 9 + 1 = 36 + 1
41 = 4 x 10 + 1 = 25 + 16
53 = 4 x 13 + 1 = 49 + 4
61 = 4 x 15 + 1 = 36 + 25
© T Madas
Prime numbers which are of the form 2 n – 1, where
n is a natural number, are called Mersenne Primes
1st Mersenne: 22 – 1 = 3
2nd Mersenne: 23 – 1 = 7
3rd Mersenne: 25 – 1 = 31
4th Mersenne: 27 – 1 = 127
5th Mersenne: 213 – 1 = 8191
6th Mersenne: 217 – 1 = 131071
7th Mersenne: 219 – 1 = 524287
8th Mersenne: 231 – 1 = 2147483647
9th Mersenne: 261 – 1 = 2305843009213693951
10th Mersenne: 289 – 1 = 618970019642690137449562111
On May 15, 2004, Josh Findley discovered the 41st
known Mersenne Prime, 224,036,583 – 1.
The number has 6 320 430 digits and is now the largest
known prime number!
© T Madas
Perfect Numbers
© T Madas
A perfect number is a number which is equal to the sum
of its factors, other than the number itself.
6 is perfect because: 1 + 2 + 3 = 6
A deficient number is a number which is more than the
sum of its factors, other than the number itself
8 is deficient because: 1 + 2 + 4 = 7
An abundant, or excessive number is a number which is
less than the sum of its factors, other than the number itself
12 is abundant because: 1 + 2 + 3 + 4 + 6 = 16
Classify the numbers from 3 to 30
according to these categories
© T Madas
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
1+2=3
1
1+2+3=6
1
1+2+4=7
1+3=4
1+2+5=8
1
1+2+3+4+6=16
1
1+2+7=10
1+3+5=9
1+2+4+8=15
D
D
D
P
D
D
D
D
D
A
D
D
D
D
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1
1+2+3+6+9=21
1
1+2+4+5+10=22
1+3+7=11
1+2+11=14
1
1+2+3+4+6+12=28
1+5=6
1+2+13=16
1+3+9=13
1+2+4+7+14=28
1
1+2+3+5+6+10+15=42
D
A
D
A
D
D
D
A
D
D
D
P
D
A
© T Madas
The definition of a perfect number dates back to the
ancient Greeks.
It was in fact Euclid that proved that a number of the form
(2n – 1)2n – 1 will be a perfect number provided that:
2n – 1 is a prime, which is known as Mersenne Prime
Since the perfect numbers are connected to the Mersenne
Primes, there are very few perfect numbers that we are
aware of, given we only know 41 Mersenne Primes
© T Madas
The definition of a perfect number dates back to the
ancient Greeks.
It was in fact Euclid that proved that a number of the form
(2n – 1)2n – 1 will be a perfect number provided that:
2n – 1 is a prime, which is known as Mersenne Prime
1st Mersenne: 22 – 1, 1st Perfect: (22 – 1)22 – 1 = 3 x 2 = 6
2nd Mersenne: 23 – 1, 2nd Perfect: (23 – 1)23 – 1 = 7 x 4 = 28
3rd Mersenne: 25 – 1, 3rd Perfect: (25 – 1)25 – 1 = 31 x 16 = 496
7
7–1
= 127 x 64 = 8128
4th Mersenne: 27 – 1, 4th Perfect: (2 – 1)2
13
13 – 1
= 33550336
5th Mersenne: 213 – 1, 5th Perfect: (2 – 1)2
© T Madas
© T Madas
Worksheets
© T Madas
© T Madas
12
22
32
42
52
62
72
82
92
11
21
31
41
51
61
71
81
91
93
83
73
63
53
43
33
23
13
3
94
84
74
64
54
44
34
24
14
4
95
85
75
65
55
45
35
25
15
5
96
86
76
66
56
46
36
26
16
6
97
87
77
67
57
47
37
27
17
7
98
88
78
68
58
48
38
28
18
8
99
89
79
69
59
49
39
29
19
9
100
90
80
70
60
50
40
30
20
10
Cross off:
number 1
multiples of 2 except 2
multiples of 3 except 3
multiples of 5 except 5
multiples of 7 except 7
Primes up to 100
2
12
22
32
42
52
62
72
82
92
1
11
21
31
41
51
61
71
81
91
93
83
73
63
53
43
33
23
13
3
94
84
74
64
54
44
34
24
14
4
95
85
75
65
55
45
35
25
15
5
96
86
76
66
56
46
36
26
16
6
97
87
77
67
57
47
37
27
17
7
98
88
78
68
58
48
38
28
18
8
99
89
79
69
59
49
39
29
19
9
100
90
80
70
60
50
40
30
20
10
Cross off:
number 1
multiples of 2 except 2
multiples of 3 except 3
multiples of 5 except 5
multiples of 7 except 7
Primes up to 100
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
2
1
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
© T Madas
12
22
32
42
52
62
72
82
92
102
112
122
132
142
152
162
172
182
192
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
193
183
173
163
153
143
133
123
113
103
93
83
73
63
53
43
33
23
13
3
194
184
174
164
154
144
134
124
114
104
94
84
74
64
54
44
34
24
14
4
Cross off:
number 1
multiples of 2 except 2
multiples of 3 except 3
multiples of 5 except 5
multiples of 7 except 7
multiples of 11 except 11
multiples of 13 except 13
Primes up to 200
2
1
195
185
175
165
155
145
135
125
115
105
95
85
75
65
55
45
35
25
15
5
196
186
176
166
156
146
136
126
116
106
96
86
76
66
56
46
36
26
16
6
197
187
177
167
157
147
137
127
117
107
97
87
77
67
57
47
37
27
17
7
198
188
178
168
158
148
138
128
118
108
98
88
78
68
58
48
38
28
18
8
199
189
179
169
159
149
139
129
119
109
99
89
79
69
59
49
39
29
19
9
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
© T Madas