# Prove that $\Phi_{420}(69) > \Phi_{69}(420)$

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.

• Nice question! ( ͡° ͜ʖ ͡°) Nov 7, 2023 at 18:37
• $68^{96}>419×421^{43}$ Nov 7, 2023 at 18:56
• 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. Nov 7, 2023 at 20:35
• Nice ( ͡° ͜ʖ ͡°) Nov 10, 2023 at 0:20