For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation} \begin{cases} x_i \in \mathbb{Z}, i=0,1,2...,n-1 \\ x_i^2 = 1, i=0,1,2...,n-1 \\ \omega = \cos(2\pi/n)+i\sin(2\pi/n) \\ z = \sum_{i=0}^{n-1} x_i \omega^{i} \\ (z*\operatorname{Conj} z ) \in \mathbb{Z} \end{cases} \end{equation}
As an example, $x_i = 1, i=0,1,2...,n-1$ is one solution to this equation. And $x_i = -1, i=0,1,2...,n-1$ is another solution. For small $n$, all solutions can be found by mathematical software. Is there any good idea for bigger $n$?
Here is the computational result for small $n$:
| $n$ | Number of solutions | $2^n$ | Percentage |
|---|---|---|---|
| $1$ | $2$ | $2$ | $100.00\%$ |
| $2$ | $4$ | $4$ | $100.00\%$ |
| $3$ | $8$ | $8$ | $100.00\%$ |
| $4$ | $16$ | $16$ | $100.00\%$ |
| $5$ | $12$ | $32$ | $37.50\%$ |
| $6$ | $64$ | $64$ | $100.00\%$ |
| $7$ | $44$ | $128$ | $34.38\%$ |
| $8$ | $144$ | $256$ | $56.25\%$ |
| $9$ | $80$ | $512$ | $15.63\%$ |
| $10$ | $244$ | $1024$ | $23.83\%$ |
| $11$ | $68$ | $2048$ | $3.32\%$ |
| $12$ | $1816$ | $4096$ | $44.34\%$ |
| $13$ | $132$ | $8192$ | $1.61\%$ |
| $14$ | $2020$ | $16384$ | $12.33\%$ |
| $15$ | $1628$ | $32768$ | $4.97\%$ |
| $16$ | $4480$ | $65536$ | $6.84\%$ |
| $17$ | $36$ | $131072$ | $0.03\%$ |
| $18$ | $17200$ | $262144$ | $6.56\%$ |
| $19$ | $116$ | $524288$ | $0.02\%$ |
| $20$ | $33416$ | $1048576$ | $3.19\%$ |
| $21$ | $6644$ | $2097152$ | $0.32\%$ |
| $22$ | $30364$ | $4194304$ | $0.72\%$ |
| $23$ | $140$ | $8388608$ | $0.00\%$ |
| $24$ | $530512$ | $16777216$ | $3.16\%$ |
| $25$ | $1032$ | $33554432$ | $0.00\%$ |
| $26$ | $173164$ | $67108864$ | $0.26\%$ |
| $27$ | $14336$ | $134217728$ | $0.01\%$ |