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I noticed that powers of complex numbers $a+bi$ tend to fall into a cycle modulo $\sqrt{a^2 + b^2}$. For example, $$(3+4i)^2 \equiv 3+4i \pmod{5}$$ and $$(-3-2\sqrt{10}i)^2 \equiv -3-2\sqrt{10}i \pmod{7}.$$

The period of the cycle could be longer as well.

Must these cycles arise for any complex number? If not, what's the justification for why the examples above having cycles?

JMP
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DozenDucc
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  • What do you mean exactly, that $\forall~a+bi\in\mathbb{C}$ there exists some positive integer $n$ such that $(a+bi)^n=(a+bi)\text{ mod }\sqrt{a^2+b^2}$? And are you sure this is true for all complex numbers? – Marcos Mar 27 '22 at 08:39
  • The result may not be true for any $a + bi$, but why is it even true for the examples given? As in, how can we justify the existence of the cycles without explicitly computing them? – DozenDucc Mar 27 '22 at 08:48
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    This seems very vague, I don't know if someone can give you a good answer, but it seems that you need to explore a lot more cases and do some work to be able to conjecture something. Having a property like this for just a bunch of random numbers is not enough to be able to think that this might be true for some families of numbers. – Marcos Mar 27 '22 at 08:53
  • Hmm, I'm working on making my question more specific. But according to this answer the cyclic behaviour should be quite general? – DozenDucc Mar 27 '22 at 09:04

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