梅森數是指形如2n−1的數, 其記號為Mn;如果一個梅森數是質(素)數它被稱為梅森質(素)數. 如n 不是質(素)數則其梅森數絕對不是梅森質(素)數. 例子請參閱下列藍色部份的證明. 如n是質(素)數則梅森數有可能是梅森質(素)數. 以下是n是質(素)數其梅森數卻不是梅森質(素)數的例子. 例如211−1 被23整除; 223−1 被47整除;2331031−1 被662063 整除. 如需更多例子請參閱下列表格.A Mersenne number is the number of the form 2n-1, the symbol for it is Mn; if a Mersenne number is prime it is called Mersenne prime. If n of Mn is not a prime, the Mersenne number definitely not a Mersenne prime. For example: If x = 123456789123456789123456789 then 2x-1 is not a prime. Proof: x is not a prime, and it is dividable by 3. Therefore, 2x-1 = (2^y)3 – 1, (that 2^y = 241152263041152263041152263 ) -> (2^y)3 – 1 = (2^y – 1) [(2^y) 2 – 2(2^y) +1], therefore, 2123456789123456789123456789-1 can be dividable by (241152263041152263041152263– 1). If n is a prime, the Mersenne number could be a prime. The following is n is prime for a Mersenne number but it is not a Mersenne prime. For example 211−1 is dividable by 23; 223−1 is dividable by 47; 2331031−1 is dividable by 662,063. For more examples please refer to the following table.
|
n是質(素)數 |
被整除的數 |
|
11 |
23 |
|
23 |
47 |
|
29 |
233 |
|
43 |
431 |
|
47 |
2351 |
|
53 |
6361 |
|
73 |
439 |
|
113 |
3391 |
|
151 |
18121 |
|
163 |
150287 |
|
173 |
730753 |
|
179 |
359 |
|
181 |
43441 |
|
223 |
18287 |
|
233 |
1399 |
|
239 |
479 |
|
251 |
503 |
|
359 |
719 |
|
397 |
2383 |
|
431 |
863 |
|
571 |
5711 |
|
641 |
35897 |
|
719 |
1439 |
|
761 |
4567 |
|
877 |
35081 |
|
883 |
8831 |
|
967 |
23209 |
|
977 |
867577 |
|
1019 |
2039 |
|
1049 |
33569 |
|
1093 |
43721 |
|
1201 |
57649 |
|
1223 |
2447 |
|
1229 |
36871 |
|
1321 |
7927 |
|
1367 |
10937 |
|
1439 |
2879 |
|
1451 |
2903 |
|
1459 |
14591 |
|
1543 |
101839 |
|
1583 |
3167 |
|
1667 |
13337 |
|
1789 |
39359 |
|
1931 |
3863 |
|
2047 |
131009 |
|
2063 |
4127 |
|
2069 |
326903 |
|
2251 |
400679 |
|
2339 |
4679 |
|
2389 |
71671 |
|
2411 |
19289 |
|
2593 |
15559 |
|
2617 |
78511 |
|
2657 |
37199 |
|
2677 |
364073 |
|
2687 |
198839 |
|
2699 |
5399 |
|
2821 |
16927 |
|
2843 |
142151 |
|
2963 |
5927 |
|
3037 |
18223 |
|
3119 |
24953 |
|
3181 |
19087 |
|
3329 |
26633 |
|
3391 |
298409 |
|
3701 |
111031 |
|
3779 |
7559 |
|
3793 |
60689 |
|
3863 |
7727 |
|
3923 |
219689 |
|
4099 |
73783 |
|
4127 |
74287 |
|
4373 |
61223 |
|
4597 |
27583 |
|
4759 |
114217 |
|
4943 |
9887 |
|
5003 |
10007 |
|
5087 |
40697 |
|
5231 |
10463 |
|
5431 |
54311 |
|
5563 |
534049 |
|
5639 |
11279 |
|
5717 |
34303 |
|
6199 |
61991 |
|
6353 |
38119 |
|
6373 |
38239 |
|
6529 |
731249 |
|
6947 |
180623 |
|
6977 |
41863 |
|
7079 |
14159 |
|
7211 |
14423 |
|
7883 |
15767 |
|
7901 |
47407 |
|
8321 |
49927 |
|
8377 |
50263 |
|
8429 |
67433 |
|
9397 |
225529 |
|
9859 |
1104209 |
|
10091 |
20183 |
|
10211 |
81689 |
|
10331 |
20663 |
|
10457 |
62743 |
|
10973 |
65839 |
|
11119 |
111191 |
|
11699 |
23399 |
|
11701 |
70207 |
|
11833 |
70999 |
|
12037 |
72223 |
|
12073 |
362191 |
|
12437 |
99497 |
|
12583 |
201329 |
|
12757 |
76543 |
|
13003 |
208049 |
|
13049 |
104393 |
|
13451 |
26903 |
|
13669 |
300719 |
|
13679 |
109433 |
|
13763 |
27527 |
|
14149 |
311279 |
|
14303 |
28607 |
|
14593 |
87559 |
|
14669 |
117353 |
|
14879 |
29759 |
|
15241 |
243857 |
|
15461 |
92767 |
|
15887 |
381289 |
|
16091 |
32183 |
|
16561 |
99367 |
|
16741 |
100447 |
|
16787 |
302167 |
|
16823 |
33647 |
|
17183 |
34367 |
|
17657 |
105943 |
|
17939 |
35879 |
|
18797 |
150377 |
|
19237 |
461689 |
|
19727 |
355087 |
|
19919 |
39839 |
|
20101 |
120607 |
|
20771 |
41543 |
|
20873 |
292223 |
|
20899 |
208991 |
|
21067 |
379207 |
|
21221 |
636631 |
|
21227 |
169817 |
|
23053 |
138319 |
|
23819 |
47639 |
|
23957 |
143743 |
|
24611 |
49223 |
|
25037 |
150223 |
|
25163 |
603913 |
|
25243 |
252431 |
|
25703 |
51407 |
|
26111 |
52223 |
|
26681 |
160087 |
|
26717 |
213737 |
|
27143 |
54287 |
|
28019 |
56039 |
|
28351 |
283511 |
|
28499 |
56999 |
|
28517 |
171103 |
|
28559 |
57119 |
|
28643 |
57287 |
|
29587 |
295871 |
|
29663 |
711913 |
|
30851 |
61703 |
|
31151 |
62303 |
|
31319 |
62639 |
|
31721 |
253769 |
|
32653 |
195919 |
|
33791 |
270329 |
|
33809 |
473327 |
|
34361 |
274889 |
|
34543 |
345431 |
|
34849 |
836377 |
|
35977 |
215863 |
|
36131 |
289049 |
|
36191 |
72383 |
|
36479 |
72959 |
|
36493 |
1094791 |
|
37957 |
227743 |
|
38231 |
76463 |
|
38237 |
229423 |
|
38459 |
76919 |
|
38461 |
230767 |
|
38611 |
694999 |
|
39209 |
548927 |
|
39779 |
79559 |
|
40559 |
81119 |
|
41603 |
83207 |
|
41959 |
419591 |
|
42719 |
85439 |
|
43391 |
86783 |
|
43691 |
87383 |
|
44699 |
89399 |
|
45179 |
90359 |
|
46073 |
276439 |
|
46199 |
92399 |
|
46601 |
279607 |
|
47363 |
94727 |
|
48337 |
290023 |
|
48353 |
290119 |
|
48731 |
97463 |
|
49919 |
99839 |
|
50411 |
100823 |
|
51647 |
929647 |
|
51659 |
103319 |
|
52571 |
105143 |
|
53353 |
320119 |
|
54779 |
438233 |
|
56099 |
112199 |
|
56923 |
2276921 |
|
58211 |
116423 |
|
58451 |
116903 |
|
59723 |
119447 |
|
60133 |
1322927 |
|
62791 |
627911 |
|
65183 |
130367 |
|
66791 |
133583 |
|
68111 |
136223 |
|
68279 |
136559 |
|
68699 |
137399 |
|
72227 |
577817 |
|
72287 |
578297 |
|
72551 |
580409 |
|
74099 |
148199 |
|
74311 |
743111 |
|
74665 |
447991 |
|
74699 |
149399 |
|
74729 |
1046207 |
|
75133 |
450799 |
|
76081 |
1217297 |
|
78779 |
157559 |
|
78781 |
472687 |
|
79087 |
790871 |
|
79151 |
158303 |
|
79193 |
475159 |
|
79367 |
634937 |
|
79811 |
159623 |
|
80369 |
642953 |
|
82193 |
493159 |
|
82609 |
1817399 |
|
82913 |
497479 |
|
83459 |
166919 |
|
85639 |
856391 |
|
85931 |
171863 |
|
86297 |
690377 |
|
87071 |
174143 |
|
87299 |
174599 |
|
90977 |
545863 |
|
94771 |
947711 |
|
96731 |
193463 |
|
96779 |
193559 |
|
98429 |
787433 |
|
98641 |
591847 |
|
100769 |
806153 |
|
101399 |
202799 |
|
102551 |
205103 |
|
103391 |
206783 |
|
104459 |
835673 |
|
105173 |
631039 |
|
105379 |
1896823 |
|
106019 |
212039 |
|
108917 |
653503 |
|
111959 |
223919 |
|
116411 |
232823 |
|
116579 |
233159 |
|
117119 |
234239 |
|
117371 |
234743 |
|
117989 |
943913 |
|
121559 |
243119 |
|
121937 |
731623 |
|
127481 |
764887 |
|
129497 |
1035977 |
|
137699 |
275399 |
|
143573 |
861439 |
|
148859 |
297719 |
|
152219 |
304439 |
|
156371 |
312743 |
|
157901 |
947407 |
|
159541 |
957247 |
|
159737 |
958423 |
|
161159 |
322319 |
|
162557 |
975343 |
|
167099 |
334199 |
|
168071 |
336143 |
|
172097 |
1032583 |
|
172541 |
1035247 |
|
182519 |
365039 |
|
182579 |
365159 |
|
184511 |
369023 |
|
191861 |
1151167 |
|
201251 |
402503 |
|
204437 |
1226623 |
|
219959 |
439919 |
|
226511 |
453023 |
|
231131 |
462263 |
|
232439 |
464879 |
|
240479 |
480959 |
|
243539 |
487079 |
|
244379 |
488759 |
|
247991 |
495983 |
|
250751 |
501503 |
|
252779 |
505559 |
|
253241 |
1519447 |
|
253751 |
507503 |
|
253919 |
507839 |
|
257561 |
1545367 |
|
259271 |
518543 |
|
265961 |
1595767 |
|
272981 |
1637887 |
|
273551 |
547103 |
|
283631 |
567263 |
|
286199 |
572399 |
|
291491 |
582983 |
|
298691 |
597383 |
|
309059 |
618119 |
|
311957 |
1871743 |
|
313037 |
1878223 |
|
317711 |
635423 |
|
322079 |
644159 |
|
331031 |
662063 |
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