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Development of
Algorithmic Constructions |
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Beschreibung
Die Primzahlen, die auf dem Polynom x^6+x+5+x^4+x^3+x^2+x+1 liegen, haben alle die Form 2*7*k+1,
bzw. alle Primzahlen, die die Form 2*7*k+1 haben, liegen auf dem Polynom x^6+x^5+x^4+x^3+x^2+x+1.
Einzige Ausnahme bildet die 7.
Die ersten Primzahlen sind:
29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449,
463, 491, 547, 617, 631, 659, 673, 701
- for x from 1 to 100 do
p:=x^6+x^5+x^4+x^3+x^2+x+1;
print (x, p, ifactor (p));
end_for;
1, 7, 7
2, 127, 127
3, 1093, 1093
4, 5461, 43 127
5, 19531, 19531
6, 55987, 55987
7, 137257, 29 4733
8, 299593, 7 127 337
9, 597871, 547 1093
10, 1111111, 239 4649
11, 1948717, 43 45319
12, 3257437, 659 4943
13, 5229043, 5229043
14, 8108731, 8108731
15, 12204241, 7 1743463
16, 17895697, 29 43 113 127
17, 25646167, 25646167
18, 36012943, 449 80207
19, 49659541, 701 70841
20, 67368421, 29 71 32719
21, 90054427, 43 631 3319
22, 118778947, 7 16968421
23, 154764793, 29 5336717
24, 199411801, 29 239 28771
25, 254313151, 29 449 19531
26, 321272407, 321272407
27, 402321277, 1093 368089
28, 499738093, 113 4422461
29, 616067011, 7 88009573
30, 754137931, 71 113 93997
31, 917087137, 917087137
32, 1108378657, 71 127 122921
33, 1331826343, 421 3163483
34, 1591616671, 463 3437617
35, 1892332261, 43 44007727
36, 2238976117, 7 29 197 55987
37, 2636996587, 71 37140797
38, 3092313043, 3092313043
39, 3611342281, 2857 1264033
40, 4201025641, 4201025641
41, 4868856847, 43 113229229
42, 5622910567, 3851 1460117
43, 6471871693, 7 5839 158341
44, 7425065341, 239 1163 26713
45, 8492487571, 29 71 4124569
46, 9684836827, 9684836827
47, 11013546097, 43 256128979
48, 12490815793, 71 175926983
49, 14129647351, 29 113 911 4733
50, 15943877551, 7 2277696793
51, 17948213557, 11411 1572887
52, 20158268677, 29 337 1303 1583
53, 22590598843, 29 778986167
54, 25262739811, 29 43 20258813
55, 28193245081, 659 42781859
56, 31401724537, 31401724537
57, 34908883807, 7 68713 72577
58, 38736564343, 211 183585613
59, 42907784221, 43 281 757 4691
60, 47446779661, 47446779661
61, 52379047267, 52379047267
62, 57731386987, 757 76263391
63, 63531945793, 6637 9572389
64, 69810262081, 7 43 127 337 5419
65, 76597310791, 29 3571 739649
66, 83925549247, 83925549247
67, 91828963717, 175897 522061
68, 100343116693, 100343116693
69, 109505194891, 3109 35221999
70, 119354057971, 20693 5767847
71, 129930287977, 7 883 21020917
72, 141276239497, 141276239497
73, 153436090543, 153436090543
74, 166455894151, 29 5739858419
75, 180383630701, 421 428464681
76, 195269260957, 195973 996409
77, 211164779827, 757 278949511
2
78, 228124270843, 7 29 43 607769
79, 246203961361, 281 337 1289 2017
80, 265462278481, 265462278481
81, 285959905687, 29 547 1093 16493
82, 307759840207, 29 10612408283
83, 330927451093, 29 11411291417
84, 355530538021, 43 2269 3643963
85, 381639390811, 7 54519912973
86, 409326849667, 379 1080018073
87, 438668366137, 438668366137
88, 469742064793, 3851 121979243
89, 502628805631, 502628805631
90, 537412247191, 43 18803 664679
91, 574178910397, 71 8087026907
92, 613018243117, 7 159293 549767
93, 654022685443, 654022685443
94, 697287735691, 29 379 63441701
95, 742912017121, 742912017121
96, 790997345377, 518813 1524629
97, 841648796647, 43 967 20241187
98, 894974776543, 239 1303 2873879
99, 951087089701, 7 12979 10468417
100, 1010101010101, 239 4649 909091
