Why does this complex number need to be simplified before computing its power?

Question

While playing with this mathematics.stackexchange.com problem I ran into an intriguing behavior. The problem is about proving that $$ x+ \frac1x=\frac{1+\sqrt5}2\Rightarrow x^{2000}+ \frac1{x^{2000}}=2. $$ The equation on the left admits $$ y=\frac{1}{4} \left(1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}}\right) $$ as a solution. A critical argument in solving the linked problem is the fact that $$ y^5=-1\text{ and therefore }y^{2000}=(-1)^{400}=1. $$ Now, asking Mathematica about all this yields

sol = Solve[x + 1/x == 1/2 (Sqrt[5] + 1), x];
y = x /. sol[[1]]
Simplify[y]

1/4 (1 + Sqrt[5] - I Sqrt[16 - (-1 - Sqrt[5])^2])
1/4 (1 + Sqrt[5] - I Sqrt[10 - 2 Sqrt[5]])

So far, so good, although I wonder why I need to invoque Simplify. However, computing $y^5$ requires Simplifying in order to obtain $-1$:

y^5
Simplify[y^5]

(1 + Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]])^5/1024
-1

and $y^{2000}$ is even more problematic

y^2000
Simplify[y^2000]
Simplify[y^5]^400

(1 + Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]])^2000/1318(...)76
(1 + Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]])^2000/1318(...)76
1

Moreover, just asking

y^2000 == Simplify[y^5]^400

does not return an answer, just

(1 + Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]])^2000/1318(...)76==1

Note that it does not appear to be related to the fact that $2000$ is large, replacing it by $20$ shows the same pattern.

My question is: what is going on? Why do I have to force simplify a complex number in order to get the correct result?

For the sake of conveinience here is my entire code:

sol = Solve[x + 1/x == 1/2 (Sqrt[5] + 1), x];
y = x /. sol[[1]]
Simplify[y]
"Testing y^5"
y^5
Simplify[y^5]
"Testing y^2000"
y^2000
Simplify[y^2000]
Simplify[y^5]^400
y^2000 == Simplify[y^5]^400

ps. I am running 11.0.1.0

Answer 1

I think the easiest way to proceed is:

{a, b} = x /. Solve[x + 1/x == (1 + Sqrt[5])/2, x];
s = ArcTan[FullSimplify[#2/#1 & @@ ReIm[a]]]
t = ArcTan[FullSimplify[#2/#1 & @@ ReIm[b]]]

Now s and t are: $-\pi/5$ and $\pi/5$ respectively and $x=e^{\pm i \pi/5}$, hence $x^{2000}=x^{-2000}=1$ and the RHS follows.

Or perhaps even simpler by noting: for $|z|=1$, $(z+1/z)/2=\cos(z)$:

z = FullSimplify[ArcCos[(1 + Sqrt[5])/4]]

yields the $z=\pi/5$ with same implication.