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# Intermediate Algebra

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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
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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
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How did you come up with those Alan?

Is there some technique you can show us?

Melody  Oct 23, 2021