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    $\bigl(\frac12(-1+\sqrt{3}i)\bigr)^n+\bigl(\frac12(-1-\sqrt{3}i)\bigr)^n$ equals $2$ if $3\mid n$ and $-1$ otherwise

    P-SW2HY

    Area
    Complex Analysis
    Topics
    Polynomials, Trigonometry
    Status
    draft

    Let \(n\) be a natural number. Show that

    \[ [1 / 2(-1+\sqrt{3} i)]^{n}+[1 / 2(-1-\sqrt{3} i)]^{n} \]

    is equal to 2 if \(n\) is a multiple of 3 , and it is equal to \(-1\) otherwise.

    Solution

    The two numbers are \[ \frac{-1+\sqrt3\,i}{2}=e^{2\pi i/3}, \qquad \frac{-1-\sqrt3\,i}{2}=e^{-2\pi i/3}. \] Therefore their \(n\)th powers sum to \[ e^{2\pi i n/3}+e^{-2\pi i n/3} =2\cos\frac{2\pi n}{3}. \] If \(3\mid n\), this equals \(2\). If \(3\nmid n\), then \(n\equiv1\) or \(2\pmod3\), so the cosine is \(-1/2\) and the sum is \(-1\). Thus the stated dichotomy holds.

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