# Aiichi Yamasaki's Homepage

## a3c

Cyclotomic polynomials $\Phi_n(x)$ are defined inductively by the following formulae. $\Phi_1(x)=x-1$ $x^m-1=\prod_{n|m} \Phi_n(x)$ here ${\displaystyle \prod_{n|m}}$ is the product over all positive divisor $n$ of $m$.

Let $n$ and $x$ be positive integers. The value $N = \Phi_n(x)$ is called a cyclotomic number. A cyclotomic polynomial $\Phi_n(x)$ is an irreducible polynomial of degree $\varphi(n)$, where $\varphi(n)$ is the Euler function. A cyclotomic number $N$ is not a prime in general.

Our goal is to factorize all the cyclotomic numbers $\Phi_n(x)$ for $\varphi(n) \leq 100, \, 2 \leq x \leq 1000.$ We have already factoreized the cyclotomic numbers $\Phi_n(x)$ for $\varphi(n) \leq 58, \, 2 \leq x \leq 1000.$

The next step is the case $\varphi(n) = 60$, i.e. $n = 61, 77, 93, 99, 122, 124, 154, 186, 198.$ Specific cyclotomic polynomials are as follows.

$\Phi_{61}$(X) = X^60 + X^59 + X^58 + X^57 + X^56 + X^55 + X^54 + X^53 + X^52 + X^51 + X^50 + X^49 + X^48 + X^47 + X^46 + X^45 + X^44 + X^43 + X^42 + X^41 + X^40 + X^39 + X^38 + X^37 + X^36 + X^35 + X^34 + X^33 + X^32 + X^31 + X^30 + X^29 + X^28 + X^27 + X^26 + X^25 + X^24 + X^23 + X^22 + X^21 + X^20 + X^19 + X^18 + X^17 + X^16 + X^15 + X^14 + X^13 + X^12 + X^11 + X^10 + X^9 + X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + X^2 + X + 1

$\Phi_{77}$(X) = X^60 - X^59 + X^53 - X^52 + X^49 - X^48 + X^46 - X^45 + X^42 - X^41 + X^39 - X^37 + X^35 - X^34 + X^32 - X^30 + X^28 - X^26 + X^25 - X^23 + X^21 - X^19 + X^18 - X^15 + X^14 - X^12 + X^11 - X^8 + X^7 - X + 1

$\Phi_{93}$(X) = X^60 - X^59 + X^57 - X^56 + X^54 - X^53 + X^51 - X^50 + X^48 - X^47 + X^45 - X^44 + X^42 - X^41 + X^39 - X^38 + X^36 - X^35 + X^33 - X^32 + X^30 - X^28 + X^27 - X^25 + X^24 - X^22 + X^21 - X^19 + X^18 - X^16 + X^15 - X^13 + X^12 - X^10 + X^9 - X^7 + X^6 - X^4 + X^3 - X + 1

$\Phi_{99}$(X) = X^60 - X^57 + X^51 - X^48 + X^42 - X^39 + X^33 - X^30 + X^27 - X^21 + X^18 - X^12 + X^9 - X^3 + 1

$\Phi_{122}$(X) = X^60 - X^59 + X^58 - X^57 + X^56 - X^55 + X^54 - X^53 + X^52 - X^51 + X^50 - X^49 + X^48 - X^47 + X^46 - X^45 + X^44 - X^43 + X^42 - X^41 + X^40 - X^39 + X^38 - X^37 + X^36 - X^35 + X^34 - X^33 + X^32 - X^31 + X^30 - X^29 + X^28 - X^27 + X^26 - X^25 + X^24 - X^23 + X^22 - X^21 + X^20 - X^19 + X^18 - X^17 + X^16 - X^15 + X^14 - X^13 + X^12 - X^11 + X^10 - X^9 + X^8 - X^7 + X^6 - X^5 + X^4 - X^3 + X^2 - X + 1

$\Phi_{124}$(X) = X^60 - X^58 + X^56 - X^54 + X^52 - X^50 + X^48 - X^46 + X^44 - X^42 + X^40 - X^38 + X^36 - X^34 + X^32 - X^30 + X^28 - X^26 + X^24 - X^22 + X^20 - X^18 + X^16 - X^14 + X^12 - X^10 + X^8 - X^6 + X^4 - X^2 + 1

$\Phi_{154}$(X) = X^60 + X^59 - X^53 - X^52 - X^49 - X^48 + X^46 + X^45 + X^42 + X^41 - X^39 + X^37 - X^35 - X^34 + X^32 - X^30 + X^28 - X^26 - X^25 + X^23 - X^21 + X^19 + X^18 + X^15 + X^14 - X^12 - X^11 - X^8 - X^7 + X + 1

$\Phi_{186}$(X) = X^60 + X^59 - X^57 - X^56 + X^54 + X^53 - X^51 - X^50 + X^48 + X^47 - X^45 - X^44 + X^42 + X^41 - X^39 - X^38 + X^36 + X^35 - X^33 - X^32 + X^30 - X^28 - X^27 + X^25 + X^24 - X^22 - X^21 + X^19 + X^18 - X^16 - X^15 + X^13 + X^12 - X^10 - X^9 + X^7 + X^6 - X^4 - X^3 + X + 1

$\Phi_{198}$(X) = X^60 + X^57 - X^51 - X^48 + X^42 + X^39 - X^33 - X^30 - X^27 + X^21 + X^18 - X^12 - X^9 + X^3 + 1

The number of digits of the integer to factorize is at most 180. Cyclotomic numbers factoring status can be found here.