# Development ofAlgorithmic Constructions

## Beschreibung

Alle Primzahlen, die die Form 4*k+1 haben, liegen auf dem Polynom x^3+x^2+x+1.
Die ersten Primzahlen dieses Typs sind:

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233,
241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401
• for x from 1 to 100 do
p:=x^3+x^2+x+1;
print (x, p, ifactor (p));
end_for;
```
2
1, 4, 2

2, 15, 3 5

3
3, 40, 2  5

4, 85, 5 17

2
5, 156, 2  3 13

6, 259, 7 37

4  2
7, 400, 2  5

2
8, 585, 3  5 13

2
9, 820, 2  5 41

10, 1111, 11 101

3
11, 1464, 2  3 61

12, 1885, 5 13 29

2
13, 2380, 2  5 7 17

14, 2955, 3 5 197

5
15, 3616, 2  113

16, 4369, 17 257

2  2
17, 5220, 2  3  5 29

2
18, 6175, 5  13 19

3
19, 7240, 2  5 181

20, 8421, 3 7 401

2
21, 9724, 2  11 13 17

22, 11155, 5 23 97

4
23, 12720, 2  3 5 53

2
24, 14425, 5  577

2
25, 16276, 2  13 313

3
26, 18279, 3  677

3
27, 20440, 2  5 7 73

28, 22765, 5 29 157

2
29, 25260, 2  3 5 421

30, 27931, 17 31 53

6
31, 30784, 2  13 37

2
32, 33825, 3 5  11 41

2
33, 37060, 2  5 17 109

34, 40495, 5 7 13 89

3  2
35, 44136, 2  3  613

36, 47989, 37 1297

2
37, 52060, 2  5 19 137

2
38, 56355, 3 5 13 17

4
39, 60880, 2  5 761

40, 65641, 41 1601

2       2
41, 70644, 2  3 7 29

42, 75895, 5 43 353

3  2
43, 81400, 2  5  11 37

2
44, 87165, 3  5 13 149

2
45, 93196, 2  23 1013

46, 99499, 29 47 73

5
47, 106080, 2  3 5 13 17

2
48, 112945, 5 7  461

2  2
49, 120100, 2  5  1201

50, 127551, 3 17 41 61

3
51, 135304, 2  13 1301

52, 143365, 5 53 541

2  3
53, 151740, 2  3  5 281

54, 160435, 5 11 2917

4
55, 169456, 2  7 17 89

56, 178809, 3 19 3137

2  3
57, 188500, 2  5  13 29

58, 198535, 5 59 673

3
59, 208920, 2  3 5 1741

60, 219661, 13 61 277

2
61, 230764, 2  31 1861

2
62, 242235, 3  5 7 769

7
63, 254080, 2  5 397

64, 266305, 5 13 17 241

2
65, 278916, 2  3 11 2113

66, 291919, 67 4357

3
67, 305320, 2  5 17 449

3
68, 319125, 3 5  23 37

2
69, 333340, 2  5 7 2381

2
70, 347971, 13  29 71

4  2
71, 363024, 2  3  2521

72, 378505, 5 17 61 73

2
73, 394420, 2  5 13 37 41

2
74, 410775, 3 5  5477

3
75, 427576, 2  19 29 97

76, 444829, 7 11 53 109

2
77, 462540, 2  3 5 13 593

78, 480715, 5 79 1217

5
79, 499360, 2  5 3121

4
80, 518481, 3  37 173

2
81, 538084, 2  17 41 193

2
82, 558175, 5  83 269

3
83, 578760, 2  3 5 7 13 53

84, 599845, 5 17 7057

2
85, 621436, 2  43 3613

86, 643539, 3 13 29 569

4
87, 666160, 2  5 11 757

88, 689305, 5 89 1549

2  2
89, 712980, 2  3  5 17 233

90, 737191, 7 13 8101

3
91, 761944, 2  23 41 101

92, 787245, 3 5 31 1693

2  2
93, 813100, 2  5  47 173

94, 839515, 5 19 8837

6
95, 866496, 2  3 4513

96, 894049, 13 97 709

2    2
97, 922180, 2  5 7  941

2
98, 950895, 3  5 11 17 113

3  2   2
99, 980200, 2  5  13  29

100, 1010101, 73 101 137

```