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Question Number 8341 by Rasheed Soomro last updated on 09/Oct/16

$$\mathrm{What}\mathrm{are}\mathrm{necessary}\mathrm{and}\mathrm{sufficient}\mathrm{conditions}$$$$\mathrm{that}{(\mathrm{a}+\mathrm{ib})}^{\mathrm{n}}\mathrm{is}\mathrm{cyclic}\mathrm{for}\mathrm{an}\mathrm{n}\mathrm{not}\mathrm{equal}\mathrm{to}0?$$

Answered by prakash jain last updated on 09/Oct/16

$$\mid a+ib\mid =1\Rightarrow \sqrt{a^2+b^2}=1$$$$\mathrm{The}\mathrm{above}\mathrm{is}\mathrm{necessary}\mathrm{and}?\mathrm{sufficient}$$$$\mathrm{condition}.$$$$\mathrm{then}\mathrm{if}$$$$arctan(b,a)=\frac{2\pi }{k}\mathrm{then}$$$${(a+bi)}^{j+mk}=(a+bi),j,m,k\in \mathbb{Z},j\in [0,k-1]$$

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