<s>[INST] If $m,n\in\mathbb{N}$ and $f(m) \mid f(n)$ for the $f$ given below, must $m=n$? If for two natural number $m$ and $n$, $(2^{2m+1}-1)2^{4m-2}(2^{2m+1}+2^{m+1}+1)\mid(2^{2n+1}-1)2^{4n-2}(2^{2n+1}+2^{n+1}+1)$, then $m=n$?
 [/INST] Another answer. Let $p(z)=(2z^2-1)(2z^2+2z+1)$.  Then you want $m<n$ and $p(2^m) \mid p(2^n)$.
Given $m$, let $D_m=p(2^m)$. Since $D_m$ is odd, if $n\equiv m \pmod {\phi(D_m)}$, then $2^n \equiv 2^m \pmod {D_m}$ and thus $D_n=p(2^n)\equiv P(2^m)=D_m \equiv 0\pmod {D_m}$
So for any $m$, there are infinitely many $n$.
Actually, you can show that $$2^{8m+4}\equiv 1\pmod {2^{2m+1}+2^{m+1}+1}$$
and
$$2^{8m+4}\equiv 1\pmod {2^{2m+1}-1}$$
So $2^{8m+4}=1\pmod {D_m}$.  So, if $m\equiv n \pmod {8m+4}$, then $p(2^m)\mid p(2^n)$.
E.g., if $m=2$, $8m+4=20$, and we get $D_2\mid D_{22}$.
More generally, if $p(z)$ is an integer polynomial with $p(0)$ odd, then for any $m$, there are infinitely many $n>m$ such that $p(2^m)\mid p(2^n)$.
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