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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 ${\displaystyle \Phi_n(x)}$ 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

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