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Find the number of pairs $\left(P(x),Q(x)\right)$ polynomial with integer coefficients, for which the identity $$P(x)^2+Q(x)^2=\left(x^{2^n}-1\right)^2$$ is hold for every positive integer $n$.

My work so far:

1) $\left(x^{2^n}-1\right)^2=(x-1)^2(x+1)^2(x^2+1)^2\cdot ...\cdot(x^{2^{n-1}+1})^2$

2) $(x^{2^k}+1)$ is a irreducible polynomial over the $\mathbb Z$ (I used by Eisenstein's criterion )

Roman83
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  • If $a$ is any $2^n$-th root of unity, we must have $P(a)^2+Q(a)^2=0$, so either $P(a)+iQ(a)=0$ or $P(a)-iQ(a)=0$. In particular, using $a=-1$ and $a=1$, we see that both $P$ and $Q$ must be divisible by $x^2-1$ in $\mathbb Z[x]$. – Ewan Delanoy Dec 05 '16 at 13:00

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