Suppose a, b, c, d are complex numbers such that

a + b + c + d = 1
a^2 + b^2 + c^2 + d^2 = 0
a^3 + b^3 + c^3 + d^3 = 1
a^4 + b^4 + c^4 + d^4 = 0

What is a^10 + b^10 + c^10 + d^10?

I can get a rough approximation using geometer's sketchpad and moving points a and b around a bit.
(c and d are reflections over the real axis = easier and imaginary parts always cancel)

a = .76 + .35i
b = -.26 + .71i

The equations come out to

1
.04
1.05
-.06

I think the solution is in the vicinity of this but moving one pixel changes the numbers rather unpredictably.

Posted by Jer on 2014-09-04 23:32:27