If x&=(-1+i√3)/(2), y&=(-1-i√3)/(2), where i^2=-1, then which of the following is not correct?

1985 AMC 12 Problem 23

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23.

\begin{aligned} x&=\frac{-1+i\sqrt3}{2},\\ y&=\frac{-1-i\sqrt3}{2}, \end{aligned}

where

i^2=-1,

then which of the following is not correct?

x^5+y^5=-1

x^7+y^7=-1

x^9+y^9=-1

x^{11}+y^{11}=-1

x^{13}+y^{13}=-1

Recognize

x

and

y

as the two nonreal cube roots of unity

Their powers depend only on the exponent modulo

3

The numbers are the two nonreal cube roots of unity, so

x^n+y^n=2\cos\left(\frac{2\pi n}{3}\right).

This equals

-1

when

n

is not divisible by

3,

but equals

2

when

n

is divisible by

3.

Of the listed exponents, only

9

is divisible by

3,

so its displayed equation is the one that is not correct.

Therefore the correct answer is C.