find-complex-number-such-that-n-m-u-v-n-m-u-v-Z-Q1498-

Question Number 1559 by 123456 last updated on 19/Aug/15

find complex number α, β such that

α^{n} = β^{m}

β^{u} = α^{v}

n, m, u, v \in Z

\boldsymbol Q 1498

Answered by Rasheed Soomro last updated on 19/Aug/15

L e t g c d (m, u) = k \Rightarrow l c m (m, u) = \frac{m u}{k}

{(α^{n})}^{\frac{u}{k}} = {(β^{m})}^{\frac{u}{k}} \Rightarrow α^{\frac{n u}{k}} = β^{\frac{m u}{k}} \dots \dots . . (\boldsymbol I)

{(β^{u})}^{\frac{m}{k}} = {(α^{v})}^{\frac{m}{k}} \Rightarrow β^{\frac{m u}{k}} = α^{\frac{m v}{k}} \dots \dots \dots (\boldsymbol I I)

[I n o r d e r t o a c h e i v e t h e c o m m o n e x p o n e n t (l e a s t a l s o) o f β]

F r o m (\boldsymbol I) a n d (\boldsymbol I I)

α^{\frac{n u}{k}} = α^{\frac{m v}{k}} \Rightarrow n u = m v [T h i s c o n d i t i o n i s r e g a r d l e s s

o f v a l u e o f α a n d β]

α^{\frac{n u}{k}} - α^{\frac{m v}{k}} = 0

α^{\frac{m v}{k}} (α^{\frac{n u}{k} - \frac{m v}{k}} - 1) = 0 \Rightarrow α^{\frac{m v}{k}} = 0 \lor α^{\frac{n u}{k} - \frac{m v}{k}} - 1 = 0

α = 0 \lor α^{\frac{n u}{k} - \frac{m v}{k}} = 1

[Y o u m a y e q u a l l y p r o c e e d a s

α^{\frac{n u}{k}} (1 - α^{\frac{m v}{k} - \frac{n u}{k}}) = 0 \Rightarrow α^{\frac{n u}{k}} = 0 \lor 1 - α^{\frac{m v}{k} - \frac{n u}{k}} = 0

\Rightarrow α = 0 \lor α^{\frac{m v}{k} - \frac{n u}{k}} = 1 \Rightarrow α^{\frac{n u}{k} - \frac{m v}{k}} = {(1)}^{- 1} = 1 s a m e a s a b o v e .]

S o α =^{\frac{n u - m v}{k}} \sqrt{1}

I - E α i s (\frac{n u - m v}{g c d (m, u)}) \boldsymbol t h r o o t o f u n i t y .

S i m i l a r p r o c e s s s h o w s t h a t

β = 0 \lor β i s (\frac{n u - m v}{g c d (n, v)}) \boldsymbol t h r o o t o f u n i t y .

S o l u t i o n f o r α

{0} \cup {1, ω, ω^{2}, \dots \dots ω^{(\frac{n u - m v}{g c d (m, u)} - 1)}}; ω i s (\frac{n u - m v}{g c d (m, u)}) \boldsymbol t h r o o t o f u n i t y .

S o l u t i o n f o r β

{0} \cup {1, μ, μ^{2}, \dots . μ^{(\frac{n u - m v}{g c d (n, v)} - 1)}}; μ i s (\frac{n u - m v}{g c d (n, v)}) \boldsymbol t h r o o t o f u n i t y .

L e t (α, β) = (ω^{p}, μ^{q}) s u c h t h a t

α^{n} = β^{m} \land β^{u} = α^{v}

{(ω^{p})}^{n} = {(μ^{q})}^{m} \land {(μ^{q})}^{u} = {(ω^{p})}^{v}

ω^{n p} = μ^{m q} \land μ^{u q} = ω^{v p}

O n s i m p l i c a t i o n w e g e t

m v = n u, W h i c h i s f r e e o f p a n d q a n d t h a t m e a n s

p a n d q m a y b e a n y i n t e g e r s .

H e n c e i f

A_{α} = {1, ω, ω^{2}, \dots \dots ω^{(\frac{n u - m v}{g c d (m, u)} - 1)}}

a n d A_{β} = {1, μ, μ^{2}, \dots . μ^{(\frac{n u - m v}{g c d (n, v)} - 1)}}, t h e s o l u t i o n f o r p a i r

(α, β) i s w h o l e A_{α} \times A_{β}

Answered by Rasheed Soomro last updated on 19/Aug/15

α a n d β m a y b e a n y c o m p l e x n u m b e r s f o r

m v = n u