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Interesting facts about prime numbers

 

 

Prime Numbers - A prime number is a whole number that only has two factors which are itself and one. For example the number 7 has only two factors which are 1 x 7 = 7. There are no other factors that you can multiply to get the answer 7. Another example would be the number 13. The only two factors that you can multiply to get 13 are 1 x 13 = 13. Here is a list of all the prime numbers between 2 and 100:

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   ( Zero and 1 are not prime or composite)

Stanislaw Ulam came up with a square spiral of consecutive integers to find out the relation amongst prime numbers. He found that all of the prime numbers fall in diagonal lines within a square spiral. The spiral starts with number 17 in the center and includes all the integers up to 272. The following table shows that the prime numbers (in blue cells) occur in diagonal lines except for a few which are by themselves.

272
271
270
269
268
267
266
265
264
263
262
261
260
259
258
257
213
212
211
210
209
208
207
206
205
204
203
202
201
200
199
256
214
161
160
159
158
157
156
155
154
153
152
151
150
149
198
255
215
162
117
116
115
114
113
112
111
110
109
108
107
148
197
254
216
163
118
81
80
79
78
77
76
75
74
73
106
147
196
253
217
164
119
82
53
52
51
50
49
48
47
72
105
146
195
252
218
165
120
83
54
33
32
31
30
29
46
71
104
145
194
251
219
166
121
84
55
34
21
20
19
28
45
70
103
144
193
250
220
167
122
85
56
35
22
17
18
27
44
69
102
143
192
249
221
168
123
86
57
36
23
24
25
26
43
68
101
142
191
248
222
169
124
87
58
37
38
39
40
41
42
67
100
141
190
247
223
170
125
88
59
60
61
62
63
64
65
66
99
140
189
246
224
171
126
89
90
91
92
93
94
95
96
97
98
139
188
245
225
172
127
128
129
130
131
132
133
134
135
136
137
138
187
244
226
173
174
175
176
177
178
179
180
181
182
183
184
185
186
243
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242

 

Another interesting fact about prime numbers is the formula developed by Leonhard Euler that produces a list of prime numbers. This formula only works for n = 0 to n = 15.

n2 + n + 17 = prime

n
n2 + n + 17 =
prime
0
02 + 0 + 17 =
17
1
12 + 1 + 17 =
19
2
22 + 2 + 17 =
23
3
32 + 3 + 17 =
29
4
42 + 4 + 17 =
37
5
52 + 5 + 17 =
47
6
62 + 6 + 17 =
59
7
72 + 7 + 17 =
73
8
82 + 8 + 17 =
89
9
92 + 9 + 17 =
107
10
102 + 10 + 17 =
127
11
112 + 11 + 17 =
149
12
122 + 12 + 17 =
173
13
132 + 13 + 17 =
199
14
142 + 14 + 17 =
227
15
152 + 15 + 17 =
257

Another formula that Euler developed to produce only prime numbers is:

n2 + n + 41 = prime

which works for values of n = 0 to n = 39. You can go ahead and try this formula.