The $n$th order cyclotomic polynomial is defined as $$ \Phi_n(x) = \prod_{\substack{1\le k\le n \\ \gcd(k, n) = 1}}{\left(x - e^{2i\pi k/n}\right)} $$ Define $c_n$ to be the smallest integer $m$ such that $\Phi_m(x)$ contains a coefficient $\pm n$. It is well known that $c_n$ contains long chains of repeated values, ($c_8 = c_9$, and $c_{10}=c_{10+i}$ for $i = 1\dots4$). Now let $C_n$ be the length of the longest chain of repeated values in $\{c_k\}_{k=1\dots n}$.
Given the definitions above, is $C_n$ unbounded?
I have no idea where to even start with this one. Any ideas?