Current Events > If x^2-x+1=0 then what is x^2025 equal to?

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x^2025=1

the first part of the challenge is to identify x
this is complicated as no real number satisfies the description so we have to go into the realm of complex numbers

I believe x = 0.5 + (square root of 3/4) i

(where i is, of course, the square root of -1)

Interesting.

Seems to me, it would have to be essentially 0. But let's crunch some numbers.

Can't factor this one (with integer coefficients, anyway), so a little quadratic formula gives us (1 +- i*root(3))/2.

I set up a spreadsheet to iterate the exponentiation, because I'm lazy like that. First row looks like this:

Iteration 1 root3 i i*root3
1 0.5 0 0 0.5

with another set of those columns next to it for the negative case. Then I just set up a formula for each column to calculate each of those coefficients when multiplied by the first row. It sounds more complicated than it was.

Then I just copied the formula down the page until I was at iteration 2025, and got -1 for either case. So x^2025 = -1. So no, it's not essentially 0 like I guessed.

I realized when I was scrolling through that the thing oscillates, and should have realized that sooner. The exponential of an imaginary oscillates, so it makes sense that the exponential of a complex might too, if you chose the coefficients well. I had a sneaking suspicion that we'd never have a coefficient of i or root3, and that turned out to be correct.

Anyway, there was definitely a less brute forcey way to do this, but whatever, my way worked, lol.

I thought this topic was about one of Elon's kids.