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In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of xn - 1 and is not a divisor of xk - 1 for any k < n. Its roots are all nth primitive roots of unity e2iπ'k/n, where k runs over the positive integers not greater than n and coprime to n (and i is the imaginary unit.

All cyclotomic polyominoes are irreducible, thus if Bunyakosky conjecture is true, then for any given integer n ≥ 1, there are infinitely many integers x ≥ 2 such that is prime.'

Cyclotomic polynomials evaluated at n[]

Number nth cyclotomic polynomial evaluated
2 2, 1, 3, 7, 5, 27, 3, X7, 15, 61, E, 1227, 11, 48X7, 37, 107, 195, 63X27, 49, 2134X7, 151, 1447, 48E, 2986627, 181, 442481, 16E7, 108001, 1X91, 12E969227, 237, 4EE2308X7, 31E15, 24XE07, 2134E, 2E0X6X7, 2401, 2277803E4X7, 85177, 3264847, 2E841, ...
3
4
5
6
7
8
9
X
E
10
11
12
13
14
15
16
17
18
19
1X
1E
20
21
22
23
24
25
26
27
28
29
2X
2E
30
31
32
33
34
35
36
37
38
39
3X
3E
40
41
42
43
44
45
46
47
48
49
4X
4E
50
51
52
53
54
55
56
57
58
59
5X
5E
60
61
62
63
64
65
66
67
68
69
6X
6E
70
71
72
73
74
75
76
77
78
79
7X
7E
80
81
82
83
84
85
86
87
88
89
8X
8E
90
91
92
93
94
95
96
97
98
99
9X
9E
X0
X1
X2
X3
X4
X5
X6
X7
X8
X9
XX
XE
E0
E1
E2
E3
E4
E5
E6
E7
E8
E9
EX
EE
100

Table of prime cyclotomic polynomials[]

Base Numbers n such that the nth cyclotomic polynomial of the base is prime
2 2, 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 12, 13, 14, 15, 17, 1X, 20, 22, 23, 26, 27, 28, 29, 2X, 32, 34, 36, 3X, 41, 48, 51, 52, 55, 59, 65, 66, 68, 71, 72, 75, 76, 79, 82, 8E, X0, X2, X6, X7, X9, E1, ...
3
4
5
6
7
8
9
X
E
10
11
12
13
14
15
16
17
18
19
1X
1E
20
21
22
23
24
25
26
27
28
29
2X
2E
30
31
32
33
34
35
36
37
38
39
3X
3E
40
41
42
43
44
45
46
47
48
49
4X
4E
50
51
52
53
54
55
56
57
58
59
5X
5E
60
61
62
63
64
65
66
67
68
69
6X
6E
70
71
72
73
74
75
76
77
78
79
7X
7E
80
81
82
83
84
85
86
87
88
89
8X
8E
90
91
92
93
94
95
96
97
98
99
9X
9E
X0
X1
X2
X3
X4
X5
X6
X7
X8
X9
XX
XE
E0
E1
E2
E3
E4
E5
E6
E7
E8
E9
EX
EE
100

See also[]