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Old 2013-02-11, 11:14   #1
paulunderwood
 
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Sep 2002
Database er0rr

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Default Mp: factors of p-1 and p+1

Has anybody looked at the factors of p-1 and p+1 if prime Mp?

Code:
? v=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609, 57885161];for(k=1,#v,p=v[k];print(p" "factor(p-1)" "factor(p+1)))
Code:
2 matrix(0,2) Mat([3, 1])
3 Mat([2, 1]) Mat([2, 2])
5 Mat([2, 2]) [2, 1; 3, 1]
7 [2, 1; 3, 1] Mat([2, 3])
13 [2, 2; 3, 1] [2, 1; 7, 1]
17 Mat([2, 4]) [2, 1; 3, 2]
19 [2, 1; 3, 2] [2, 2; 5, 1]
31 [2, 1; 3, 1; 5, 1] Mat([2, 5])
61 [2, 2; 3, 1; 5, 1] [2, 1; 31, 1]
89 [2, 3; 11, 1] [2, 1; 3, 2; 5, 1]
107 [2, 1; 53, 1] [2, 2; 3, 3]
127 [2, 1; 3, 2; 7, 1] Mat([2, 7])
521 [2, 3; 5, 1; 13, 1] [2, 1; 3, 2; 29, 1]
607 [2, 1; 3, 1; 101, 1] [2, 5; 19, 1]
1279 [2, 1; 3, 2; 71, 1] [2, 8; 5, 1]
2203 [2, 1; 3, 1; 367, 1] [2, 2; 19, 1; 29, 1]
2281 [2, 3; 3, 1; 5, 1; 19, 1] [2, 1; 7, 1; 163, 1]
3217 [2, 4; 3, 1; 67, 1] [2, 1; 1609, 1]
4253 [2, 2; 1063, 1] [2, 1; 3, 1; 709, 1]
4423 [2, 1; 3, 1; 11, 1; 67, 1] [2, 3; 7, 1; 79, 1]
9689 [2, 3; 7, 1; 173, 1] [2, 1; 3, 1; 5, 1; 17, 1; 19, 1]
9941 [2, 2; 5, 1; 7, 1; 71, 1] [2, 1; 3, 1; 1657, 1]
11213 [2, 2; 2803, 1] [2, 1; 3, 2; 7, 1; 89, 1]
19937 [2, 5; 7, 1; 89, 1] [2, 1; 3, 1; 3323, 1]
21701 [2, 2; 5, 2; 7, 1; 31, 1] [2, 1; 3, 1; 3617, 1]
23209 [2, 3; 3, 1; 967, 1] [2, 1; 5, 1; 11, 1; 211, 1]
44497 [2, 4; 3, 3; 103, 1] [2, 1; 19, 1; 1171, 1]
86243 [2, 1; 13, 1; 31, 1; 107, 1] [2, 2; 3, 1; 7187, 1]
110503 [2, 1; 3, 2; 7, 1; 877, 1] [2, 3; 19, 1; 727, 1]
132049 [2, 4; 3, 2; 7, 1; 131, 1] [2, 1; 5, 2; 19, 1; 139, 1]
216091 [2, 1; 3, 2; 5, 1; 7, 4] [2, 2; 89, 1; 607, 1]
756839 [2, 1; 23, 1; 16453, 1] [2, 3; 3, 1; 5, 1; 7, 1; 17, 1; 53, 1]
859433 [2, 3; 7, 1; 103, 1; 149, 1] [2, 1; 3, 1; 143239, 1]
1257787 [2, 1; 3, 2; 69877, 1] [2, 2; 7, 1; 29, 1; 1549, 1]
1398269 [2, 2; 349567, 1] [2, 1; 3, 1; 5, 1; 127, 1; 367, 1]
2976221 [2, 2; 5, 1; 13, 1; 11447, 1] [2, 1; 3, 1; 401, 1; 1237, 1]
3021377 [2, 6; 17, 1; 2777, 1] [2, 1; 3, 1; 503563, 1]
6972593 [2, 4; 11, 1; 173, 1; 229, 1] [2, 1; 3, 1; 1162099, 1]
13466917 [2, 2; 3, 2; 83, 1; 4507, 1] [2, 1; 149, 1; 45191, 1]
20996011 [2, 1; 3, 4; 5, 1; 7, 2; 23, 2] [2, 2; 83, 1; 63241, 1]
24036583 [2, 1; 3, 1; 4006097, 1] [2, 3; 11, 1; 13, 1; 21011, 1]
25964951 [2, 1; 5, 2; 11, 1; 17, 1; 2777, 1] [2, 3; 3, 1; 13, 1; 83221, 1]
30402457 [2, 3; 3, 1; 7, 1; 37, 1; 67, 1; 73, 1] [2, 1; 23, 1; 660923, 1]
32582657 [2, 10; 47, 1; 677, 1] [2, 1; 3, 1; 5430443, 1]
37156667 [2, 1; 19, 1; 59, 1; 16573, 1] [2, 2; 3, 1; 3096389, 1]
43112609 [2, 5; 7, 1; 11, 1; 17497, 1] [2, 1; 3, 2; 5, 1; 479029, 1]
57885161 [2, 3; 5, 1; 29, 1; 139, 1; 359, 1] [2, 1; 3, 1; 9647527, 1]
p+1=6*q or 12*q , q prime, turns up a lot. If I had spare computing cycles I would concentrate on this type with p-1 divisible by high powers of 2.
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