If $ω$ is the imaginary cube root of unity, then find the number of pairs of integers (a, b) such that $| a ω + b | = 1$ .

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Solution

$ω = \frac{- 1 + \sqrt{3} i}{2}$
put value of $ω$

$| a (\frac{- 1 + \sqrt{3} i}{2}) + b | = 1$

simplify this and get

$a^{2} + b^{2} - a b = 1$

total 4 pairs $(0, 1), (1, 0), (- 1, 0), (0, - 1)$

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