The other day my friend was asked to find $A$ and $B$ in the equation $$(x^3+2x+1)^{17} \equiv Ax+B \pmod {x^2+1}$$ A method was proposed by our teacher to use complex numbers and especially to let $x=i$ where $i$ is the imaginary unit. We obtain from that substitution $$(i+1)^{17} \equiv Ai+B \pmod {0}$$ which if we have understood it correctly is valid if we define $a \equiv b \pmod n$ to be $a=b+dn$. Running through with this definition we have $$\begin{align*} (i+1)^{17} &=\left(\sqrt{2}\left(\cos\left(\frac{\pi}{4}\right)+\sin\left(\frac{\pi}{4}\right)\right)\right)^{17}\\ &=\sqrt{2}^{17}\left(\cos\left(\frac{17\pi}{4}\right)+\sin\left(\frac{17\pi}{4}\right)\right) \tag{De Moivre}\\ &=256\left(\sqrt{2}\left(\cos\left(\frac{\pi}{4}\right)+\sin\left(\frac{\pi}{4}\right)\right)\right)\\ &=256\left(1+i\right) \\ &=256+256i\end{align*} $$ which gives the correct coefficient values for $A$ and $B$. Our questions are
- Why is this substitution valid to begin with?
- It seems here that the special case ($x=i$) implies the general case ($x$), why is that valid?
from Hot Weekly Questions - Mathematics Stack Exchange
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