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Question Number 146863 by lyubita last updated on 16/Jul/21

Commented by MJS_new last updated on 16/Jul/21

$q u e s t i o n 144084$
	Answered by Olaf_Thorendsen last updated on 16/Jul/21

	$a + \frac{1}{a} = - 1$ $a^{2} + a + 1 = 0$ $a = \frac{- 1 \pm i \sqrt{3}}{2} = e^{\frac{2 i π}{3}} or e^{\frac{4 i π}{3}}$ $1 st case : a = e^{\frac{2 i π}{3}}$ $n = 1234567891011 = 3 k, k \in Z$ $because 1 + 2 + 3 + 4 + . . . + 1 + 0 + 1 + 1 = 48 = 3 p$ $\Rightarrow a^{n} = {(e^{\frac{2 i π}{3}})}^{3 k} = e^{2 i k π} = 1$ $m = 1110987654321 = 3 k^{'}, k^{'} \in Z$ $because 1 + 1 + 1 + 0 + . . . + 4 + 3 + 2 + 1 = 48 = 3 p$ $\frac{1}{a^{m}} = {(e^{- \frac{2 i π}{3}})}^{3 k^{'}} = e^{- 2 {i k}^{'} π} = 1$ $\Rightarrow a^{n} + \frac{1}{a^{m}} = 1 + 1 = 2$ $Same result with a = e^{\frac{4 i π}{3}}$
	Commented by lyubita last updated on 16/Jul/21

	$t h a n k y o u s o m u c h . a r e y o u f r o m f r e n c h ?$
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