+0  
 
0
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Answer with explanation please. Thanks

 

 

Let \(\omega\) be a nonreal root of \(z^3 = 1.\) Find the number of ordered pairs \((a,b)\) of integers such that \(|a \omega + b| = 1.\)

 Oct 22, 2021
 #1
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There are four pairs that work: (1,0), (-1,0), (0,1), (0,-1).

 Oct 22, 2021
 #2
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They are the only ones I can think of too.

 

I got down to

 

\(a^2+b^2 -ab=1\)

 

Not sure how to determine if those are the only 4 answers.

 Oct 23, 2021
 #3
avatar+32738 
+4

The following also work:

  \(a=\frac{2}{\sqrt3}, b=\frac{1}{\sqrt3}\\a=-\frac{2}{\sqrt3}, b = -\frac{1}{\sqrt3}\)

Alan  Oct 23, 2021
 #4
avatar+115340 
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How did you come up with those Alan?  

Is there some technique you can show us?

Melody  Oct 23, 2021

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