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Let $\Phi_n(x)$ denote the nth cyclotomic polynomial. Prove that $\Phi_{420}(69) > \Phi_{69}(420)$.

Observe that if $\phi$ denotes the Euler-phi function, \begin{align} \phi(420) &= (2^2-2) \cdot (5-1) \cdot (3-1)\cdot (7-1) = 96, \\ \phi(69) &= (3-1) \cdot (23-1) = 44. \end{align} We want to show that $$ (69^{420} - 1) \hspace{-1em} \prod_{\substack{0\leq k < 69 \\ \gcd(k,69) > 1}} \hspace{-1em} (420-e^{2 i \pi k/69}) \; > \; (420^{69} - 1) \hspace{-1em} \prod_{\substack{0\leq k<420, \\ \gcd(k,420) > 1}} \hspace{-1em} (69-e^{2 i \pi k/420}). $$ But both sides of the inequality we need to show look very complicated, and I'm not sure how to simplify them. The product $$ \prod_{\substack{0\leq k<420, \\ \gcd(k,420) > 1}} \hspace{-1em} (e^{2 i \pi k/420}) $$ includes $210$th, $84$th, $140$th, and $60$th roots of unity. Note that an $a$th root of unity that is also a $b$th root of unity is a $\gcd(a,b)$th root of unity, since $\gcd(a,b)$ is a linear combination of a and b.

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    $\begingroup$ Nice question! ( ͡° ͜ʖ ͡°) $\endgroup$ Nov 7, 2023 at 18:37
  • $\begingroup$ $68^{96}>419×421^{43}$ $\endgroup$
    – Bob Dobbs
    Nov 7, 2023 at 18:56
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    $\begingroup$ It's easier to use the definition eg here, that $\Phi_n(x)$ is the product of $(x - e^{2i \pi k/n})$ for $k$ coprime to $n$. As Bob remarks, this means $\Phi_{420}(69)$ is a product of $96$ complex numbers which are all close to $69$ and $\Phi_{69}(420)$ is a product of $44$ numbers which are all close to $420$. You can use this to obtain upper and lower bounds. $\endgroup$ Nov 7, 2023 at 20:35
  • $\begingroup$ Nice ( ͡° ͜ʖ ͡°) $\endgroup$ Nov 10, 2023 at 0:20

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