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Old 2013-02-13, 06:47   #7
paulunderwood
 
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Sep 2002
Database er0rr

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Some more speculation; this time about p^2-2.

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^2-2)))
Code:
2 Mat([2, 1])
3 Mat([7, 1])
5 Mat([23, 1])
7 Mat([47, 1])
13 Mat([167, 1])
17 [7, 1; 41, 1]
19 Mat([359, 1])
31 [7, 1; 137, 1]
61 Mat([3719, 1])
89 Mat([7919, 1])
107 Mat([11447, 1])
127 Mat([16127, 1])
521 [7, 1; 17, 1; 2281, 1]
607 Mat([368447, 1])
1279 [31, 1; 52769, 1]
2203 [23, 1; 79, 1; 2671, 1]
2281 [367, 1; 14177, 1]
3217 [7, 1; 151, 1; 9791, 1]
4253 [7, 2; 369143, 1]
4423 Mat([19562927, 1])
9689 [47, 1; 1063, 1; 1879, 1]
9941 [23, 1; 4296673, 1]
11213 [521, 1; 241327, 1]
19937 [9551, 1; 41617, 1]
21701 [127, 1; 3708137, 1]
23209 [7, 1; 76951097, 1]
44497 [359, 1; 5515273, 1]
86243 [7, 1; 1062550721, 1]
110503 [503, 1; 3391, 1; 7159, 1]
132049 [1223, 1; 14257513, 1]
216091 [109103, 1; 427993, 1]
756839 [153841, 1; 3723359, 1]
859433 Mat([738625081487, 1])
1257787 [1217, 1; 5903, 1; 220217, 1]
1398269 [97, 1; 6703, 1; 3007049, 1]
2976221 [7, 2; 47, 1; 1447, 1; 2658079, 1]
3021377 [23, 1; 167, 1; 2376651647, 1]
6972593 [23, 2; 116959, 1; 785777, 1]
13466917 [47, 1; 3858677733721, 1]
20996011 [233, 1; 1891984883743, 1]
24036583 [7, 1; 17, 1; 4855103548873, 1]
25964951 Mat([674178680432399, 1])
30402457 Mat([924309391636847, 1])
32582657 [257, 1; 1583, 1; 2609509937, 1]
37156667 [73, 1; 2687, 1; 7038546337, 1]
43112609 Mat([1858697054786879, 1])
57885161 [31, 1; 313, 1; 48889, 1; 7063457, 1]
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