I use 1 because though it is not prime, it is relatively prime to every positive integer (including itself).
If you think of this like a sieve:
30k+ |
1 |
|
|
|
|
|
7 |
|
|
|
11 |
|
13 |
|
|
|
17 |
|
19 |
|
|
|
23 |
|
|
|
|
|
29 |
0 |
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 |
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 |
201 |
202 |
203 |
204 |
205 |
206 |
207 |
208 |
209 |
210 |
211 |
212 |
213 |
214 |
215 |
216 |
217 |
218 |
219 |
220 |
221 |
222 |
223 |
224 |
225 |
226 |
227 |
228 |
229 |
230 |
231 |
232 |
233 |
234 |
235 |
236 |
237 |
238 |
239 |
240 |
241 |
242 |
243 |
244 |
245 |
246 |
247 |
248 |
249 |
250 |
251 |
252 |
253 |
254 |
255 |
256 |
257 |
258 |
259 |
260 |
261 |
262 |
263 |
264 |
265 |
266 |
267 |
268 |
269 |
270 |
271 |
272 |
273 |
274 |
275 |
276 |
277 |
278 |
279 |
280 |
281 |
282 |
283 |
284 |
285 |
286 |
287 |
288 |
289 |
290 |
291 |
292 |
293 |
294 |
295 |
296 |
297 |
298 |
299 |
300 |
301 |
302 |
303 |
304 |
305 |
306 |
307 |
308 |
309 |
310 |
311 |
312 |
313 |
314 |
315 |
316 |
317 |
318 |
319 |
320 |
321 |
322 |
323 |
324 |
325 |
326 |
327 |
328 |
329 |
330 |
331 |
332 |
333 |
334 |
335 |
336 |
337 |
338 |
339 |
340 |
341 |
342 |
343 |
344 |
345 |
346 |
347 |
348 |
349 |
350 |
351 |
352 |
353 |
354 |
355 |
356 |
357 |
358 |
359 |
360 |
361 |
362 |
363 |
364 |
365 |
366 |
367 |
368 |
369 |
370 |
371 |
372 |
373 |
374 |
375 |
376 |
377 |
378 |
379 |
380 |
381 |
382 |
383 |
384 |
385 |
386 |
387 |
388 |
389 |
390 |
391 |
392 |
393 |
394 |
395 |
396 |
397 |
398 |
399 |
400 |
401 |
402 |
403 |
404 |
405 |
406 |
407 |
408 |
409 |
410 |
411 |
412 |
413 |
414 |
415 |
416 |
417 |
418 |
419 |
420 |
421 |
422 |
423 |
424 |
425 |
426 |
427 |
428 |
429 |
430 |
431 |
432 |
433 |
434 |
435 |
436 |
437 |
438 |
439 |
440 |
441 |
442 |
443 |
444 |
445 |
446 |
447 |
448 |
449 |
450 |
451 |
452 |
453 |
454 |
455 |
456 |
457 |
458 |
459 |
460 |
461 |
462 |
463 |
464 |
465 |
466 |
467 |
468 |
469 |
470 |
471 |
472 |
473 |
474 |
475 |
476 |
477 |
478 |
479 |
All of the columns lead with a number (these are 1,7,11,13,17,19,23,29) are the ones that contain all of the primes plus some composites. Any column headed by a blank means that they all share a similar non-trivial factor with 30. For example, every second column contains even numbers, every third contains numbers divisible by three, every 5th, 6th, 10th, and 15th work similarly. These also grab the 4th, 8th, 9th, 12th, 14th, 16th, 18th, 20th, 21st, 25th, and 27th rows for removal.
With this visual, I think you can see how it can determine if a number is definitely composite or possibly prime. As you go to larger and larger numbers, these fixed sieves become less and less 'sure' about the primality of a number. To get an idea of which composites won't be removed from the sieve above, they will be all the composites whose smallest prime factor is 7 or larger, starting at 49 (so it definitely gives you everything less than 49— not much :P). 49,77,91,119,133, 161, 203,… These are all divisible by 7 and won't be removed by the sieve.
Also, sorry for the large table :/ It seemed like the easiest way to show it.