Math Question and Answers


Question and Answers Forum

All Questions Topic List
Number Theory Questions
Previous in All Question Next in All Question
Previous in Number Theory Next in Number Theory
Question Number 134327 by mr W last updated on 02/Mar/21

	Answered by aleks041103 last updated on 25/Dec/21

	$i \Rightarrow∣ z_{k}^{2} ∣= 1$ $\Rightarrow z_{k} = e^{{i t}_{k}}$ $w_{k} = z_{k}^{2} = e^{2 {i t}_{k}}$ $\Rightarrow w_{1} + w_{2} + w_{3} = 0$ $\Rightarrow 1 + e^{2 i (t_{2} - t_{1})} + e^{2 i (t_{3} - t_{1})} = 0$ $\Rightarrow I m (e^{2 i (t_{2} - t_{1})}) = - I m (e^{2 i (t_{3} - t_{1})})$ $\Rightarrow t_{2} - t_{1} = t_{1} - t_{3}$ $a l s o$ $2 (t_{2} - t_{1}) = \frac{2 π}{3} + 2 π m$ $\Rightarrow t_{2} - t_{1} = \frac{π}{3} + m π$ $\Rightarrow t_{3} - t_{1} = p π - \frac{π}{3}$ $\Rightarrow z_{2} = z_{1} e^{i (t_{2} - t_{1})} = z_{1} e^{i \frac{π}{3}} e^{i m π} = \pm z_{1} e^{\frac{i π}{3}}$ $z_{3} = z_{1} e^{i (t_{3} - t_{1})} = z_{1} e^{- \frac{i π}{3}} e^{i p π} = \pm z_{1} e^{- \frac{i π}{3}}$ $z_{1} + z_{2} + z_{3} =$ $= z_{1} (1 \pm e^{\frac{i π}{3}} \pm e^{- \frac{i π}{3}}) \neq 0$ $\Rightarrow 1 + s_{1} (\frac{1}{2} + i \frac{\sqrt{3}}{2}) + s_{2} (\frac{1}{2} - i \frac{\sqrt{3}}{2}) \neq 0$ $i f - 2 = s_{1} + s_{2} a n d s_{1} - s_{2} = 0 t h e n z_{1} + z_{2} + z_{3} = 0$ $o n l y w h e n s_{1} = s_{2} = - 1$ $C a s e 1 :$ $z_{1}, z_{2} = - z_{1} e^{i π / 3} = z_{1} e^{4 i π / 3}, z_{3} = z_{1} e^{- i π / 3}$ $∣ z_{1}^{n} + z_{2}^{n} + z_{3}^{n} ∣=∣ z_{1} ∣^{n} ∣ 1 + e^{4 n i π / 3} + e^{- n i π / 3} ∣=$ $=∣ 1 + e^{4 n i π / 3} + e^{- n i π / 3} ∣$ $. . . .$ $t h e r e s t i s t r i v i a l$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum