A prime p such that 2p + 1 is also prime. more
The first 600 Sophie Germain primes :
2,
3,
5,
11,
23,
29,
41,
53,
83,
89,
113,
131,
173,
179,
191,
233,
239,
251,
281,
293,
359,
419,
431,
443,
491,
509,
593,
641,
653,
659,
683,
719,
743,
761,
809,
911,
953,
1013,
1019,
1031,
1049,
1103,
1223,
1229,
1289,
1409,
1439,
1451,
1481,
1499,
1511,
1559,
1583,
1601,
1733,
1811,
1889,
1901,
1931,
1973,
2003,
2039,
2063,
2069,
2129,
2141,
2273,
2339,
2351,
2393,
2399,
2459,
2543,
2549,
2693,
2699,
2741,
2753,
2819,
2903,
2939,
2963,
2969,
3023,
3299,
3329,
3359,
3389,
3413,
3449,
3491,
3539,
3593,
3623,
3761,
3779,
3803,
3821,
3851,
3863,
3911,
4019,
4073,
4211,
4271,
4349,
4373,
4391,
4409,
4481,
4733,
4793,
4871,
4919,
4943,
5003,
5039,
5051,
5081,
5171,
5231,
5279,
5303,
5333,
5399,
5441,
5501,
5639,
5711,
5741,
5849,
5903,
6053,
6101,
6113,
6131,
6173,
6263,
6269,
6323,
6329,
6449,
6491,
6521,
6551,
6563,
6581,
6761,
6899,
6983,
7043,
7079,
7103,
7121,
7151,
7193,
7211,
7349,
7433,
7541,
7643,
7649,
7691,
7823,
7841,
7883,
7901,
8069,
8093,
8111,
8243,
8273,
8513,
8663,
8693,
8741,
8951,
8969,
9029,
9059,
9221,
9293,
9371,
9419,
9473,
9479,
9539,
9629,
9689,
9791,
10061,
10091,
10163,
10253,
10271,
10313,
10331,
10529,
10589,
10613,
10691,
10709,
10733,
10781,
10799,
10883,
11171,
11321,
11369,
11393,
11471,
11519,
11549,
11579,
11699,
11783,
11801,
11813,
11831,
11909,
11939,
12011,
12041,
12101,
12119,
12203,
12263,
12329,
12653,
12671,
12791,
12821,
12899,
12923,
12959,
13001,
13049,
13229,
13313,
13451,
13463,
13553,
13619,
13649,
13763,
13883,
13901,
13913,
14009,
14081,
14153,
14159,
14249,
14303,
14321,
14489,
14561,
14621,
14669,
14699,
14741,
14783,
14831,
14879,
14939,
15101,
15161,
15173,
15233,
15269,
15401,
15569,
15629,
15773,
15791,
15803,
15923,
16001,
16091,
16253,
16301,
16421,
16493,
16553,
16673,
16811,
16823,
16883,
16931,
17159,
17183,
17291,
17333,
17351,
17579,
17669,
17681,
17939,
17981,
18041,
18131,
18149,
18191,
18233,
18341,
18443,
18461,
18731,
18773,
18803,
18899,
19163,
19301,
19319,
19373,
19391,
19433,
19553,
19559,
19661,
19709,
19751,
19889,
19913,
19919,
19991,
20063,
20249,
20369,
20393,
20411,
20441,
20693,
20753,
20759,
20771,
20789,
20879,
20921,
20939,
20963,
21011,
21089,
21149,
21179,
21221,
21341,
21383,
21419,
21611,
21701,
21713,
21803,
21893,
22013,
22079,
22133,
22259,
22271,
22343,
22349,
22409,
22433,
22469,
22481,
22541,
22751,
22853,
22943,
23099,
23279,
23321,
23339,
23459,
23561,
23603,
23669,
23753,
23819,
23909,
23981,
24203,
24239,
24281,
24473,
24509,
24551,
24611,
24683,
24749,
24971,
25073,
25229,
25523,
25601,
25643,
25673,
25703,
25799,
25841,
25913,
25919,
26111,
26189,
26459,
26501,
26573,
26633,
26849,
26879,
26891,
26993,
27143,
27281,
27479,
27539,
27551,
27581,
27743,
27773,
27809,
27893,
27983,
28001,
28019,
28403,
28499,
28559,
28571,
28643,
28751,
28793,
28859,
28901,
28949,
28961,
29021,
29033,
29201,
29339,
29363,
29453,
29483,
29531,
29723,
29873,
30269,
30323,
30389,
30449,
30671,
30689,
30773,
30851,
30983,
31019,
31151,
31253,
31319,
31469,
31649,
31721,
31793,
31799,
31859,
32003,
32009,
32141,
32159,
32381,
32531,
32561,
32573,
32633,
32771,
32789,
32843,
32933,
33023,
33053,
33119,
33179,
33191,
33461,
33479,
33521,
33569,
33623,
33713,
33749,
33773,
33809,
33941,
34253,
34283,
34319,
34439,
34631,
34883,
34913,
34949,
35069,
35081,
35099,
35111,
35291,
35573,
35831,
35933,
35993,
35999,
36083,
36191,
36251,
36353,
36383,
36479,
36563,
36629,
36761,
36791,
36821,
36923,
36929,
37013,
37049,
37139,
37181,
37253,
37379,
37619,
37853,
37871,
37991,
38039,
38183,
38189,
38201,
38231,
38303,
38333,
38453,
38459,
38501,
38639,
38669,
38723,
38861,
38873,
38891,
38933,
39089,
39233,
39239,
39419,
39443,
39521,
39551,
39569,
39659,
39779,
39953,
39971,
39983,
39989,
40193,
40283,
40343,
40559,
40763,
40823,
40853,
40949,
41081,
41231,
41243,
41381,
41399,
41603,
41609,
41621,
41669,
41729,
41969,
42023,
42071,
42089,
42131,
42221,
42359,
42473,
42611,
42719,
42743,
42821,
42923,
43013,
43313,
43391,
43541.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 26569516 values, from 2 to 10000000343).
n\r | 0 | 1 |
2 | 1 | 26569515 | 2 |
3 | 1 | 0 | 26569515 | 3 |
4 | 0 | 13283194 | 1 | 13286321 | 4 |
5 | 1 | 8856045 | 1 | 8858802 | 8854667 | 5 |
6 | 0 | 0 | 1 | 1 | 0 | 26569514 | 6 |
7 | 0 | 5312442 | 5314898 | 1 | 5316337 | 5311455 | 5314383 | 7 |
8 | 0 | 6643473 | 1 | 6641754 | 0 | 6639721 | 0 | 6644567 | 8 |
9 | 0 | 0 | 8855992 | 1 | 0 | 8855532 | 0 | 0 | 8857991 | 9 |
10 | 0 | 8856045 | 1 | 8858802 | 0 | 1 | 0 | 0 | 0 | 8854667 | 10 |
11 | 1 | 2952313 | 2953614 | 2952518 | 2951887 | 1 | 2951044 | 2952376 | 2952239 | 2952284 | 2951239 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.