For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation} \begin{array}l x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\ x_i^2 = 1, i=0,1,2\dotsc,n-1 \\ \omega = \cos(2\pi/n)+i\sin(2\pi/n) \\ z = \sum_{i=0}^{n-1} x_i \omega^{i} \\ \lvert z\rvert^2 \in \mathbb{Z}. \end{array} \end{equation}
As an example, $x_i = 1$, $i=0,1,2\dotsc,n-1$ is one solution to this equation. And $x_i = -1$, $i=0,1,2\dotsc,n-1$ 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% |
2 | 4 | 4 | 100% |
3 | 8 | 8 | 100% |
4 | 16 | 16 | 100% |
5 | 12 | 32 | 37.5% |
6 | 64 | 64 | 100% |
7 | 44 | 128 | 34.375% |
8 | 144 | 256 | 56.25% |
9 | 80 | 512 | 15.625% |
10 | 244 | 1024 | 23.8281% |
11 | 68 | 2048 | 3.32031% |
12 | 1816 | 4096 | 44.3359% |
13 | 132 | 8192 | 1.61132% |
14 | 2020 | 16384 | 12.3291% |
15 | 1628 | 32768 | 4.96826% |
16 | 4480 | 65536 | 6.83593% |
17 | 36 | 131072 | 0.02746% |
18 | 17200 | 262144 | 6.56127% |
19 | 116 | 524288 | 0.02212% |
20 | 33416 | 1048576 | 3.18679% |
21 | 6644 | 2097152 | 0.31681% |
22 | 30364 | 4194304 | 0.72393% |
23 | 140 | 8388608 | 0.00166% |
24 | 530512 | 16777216 | 3.16209% |
25 | 832 | 33554432 | 0.00247% |
26 | 173164 | 67108864 | 0.25803% |
27 | 14336 | 134217728 | 0.01068% |
28 | 673024 | 268435456 | 0.25072% |
29 | 60 | 536870912 | 0.00001% |
30 | 12263284 | 1073741824 | 1.14210% |
Here is the further detail, the prime number related patterns are quite obvious.
$n$ | $\lvert z\rvert^2$ | Number of solutions |
---|---|---|
1 | 1 | 2 |
2 | 0 | 2 |
2 | 4 | 2 |
3 | 0 | 2 |
3 | 4 | 6 |
4 | 0 | 4 |
4 | 4 | 8 |
4 | 8 | 4 |
5 | 0 | 2 |
5 | 4 | 10 |
6 | 0 | 10 |
6 | 4 | 36 |
6 | 12 | 12 |
6 | 16 | 6 |
7 | 0 | 2 |
7 | 4 | 14 |
7 | 8 | 28 |
8 | 0 | 16 |
8 | 4 | 64 |
8 | 8 | 32 |
8 | 12 | 32 |
9 | 0 | 8 |
9 | 4 | 72 |
10 | 0 | 34 |
10 | 4 | 180 |
10 | 16 | 10 |
10 | 20 | 20 |
11 | 0 | 2 |
11 | 4 | 22 |
11 | 12 | 44 |
12 | 0 | 100 |
12 | 4 | 720 |
12 | 8 | 432 |
12 | 12 | 240 |
12 | 16 | 120 |
12 | 20 | 144 |
12 | 24 | 48 |
12 | 32 | 12 |
13 | 0 | 2 |
13 | 4 | 26 |
13 | 12 | 104 |
14 | 0 | 130 |
14 | 4 | 924 |
14 | 8 | 672 |
14 | 16 | 238 |
14 | 28 | 28 |
14 | 32 | 28 |
15 | 0 | 38 |
15 | 4 | 600 |
15 | 8 | 600 |
15 | 12 | 60 |
15 | 16 | 210 |
15 | 20 | 60 |
15 | 24 | 60 |
16 | 0 | 256 |
16 | 4 | 2048 |
16 | 8 | 1024 |
16 | 12 | 1024 |
16 | 28 | 128 |
17 | 0 | 2 |
17 | 4 | 34 |
18 | 0 | 1000 |
18 | 4 | 10800 |
18 | 12 | 3600 |
18 | 16 | 1800 |
19 | 0 | 2 |
19 | 4 | 38 |
19 | 20 | 76 |
20 | 0 | 1156 |
20 | 4 | 12240 |
20 | 8 | 6480 |
20 | 12 | 5760 |
20 | 16 | 680 |
20 | 20 | 4640 |
20 | 24 | 1440 |
20 | 28 | 640 |
20 | 32 | 20 |
20 | 36 | 80 |
20 | 40 | 240 |
20 | 48 | 40 |
21 | 0 | 134 |
21 | 4 | 2856 |
21 | 8 | 2184 |
21 | 12 | 84 |
21 | 16 | 714 |
21 | 24 | 168 |
21 | 28 | 420 |
21 | 32 | 84 |
22 | 0 | 2050 |
22 | 4 | 22572 |
22 | 12 | 4224 |
22 | 16 | 22 |
22 | 20 | 1408 |
22 | 44 | 44 |
22 | 48 | 44 |
23 | 0 | 2 |
23 | 4 | 46 |
23 | 24 | 92 |
24 | 0 | 10000 |
24 | 4 | 144000 |
24 | 8 | 86400 |
24 | 12 | 151680 |
24 | 16 | 24000 |
24 | 20 | 63360 |
24 | 24 | 26880 |
24 | 28 | 11520 |
24 | 32 | 2400 |
24 | 36 | 6720 |
24 | 40 | 1920 |
24 | 44 | 960 |
24 | 48 | 480 |
24 | 60 | 192 |
25 | 0 | 32 |
25 | 4 | 800 |
26 | 0 | 8194 |
26 | 4 | 106548 |
26 | 12 | 54912 |
26 | 16 | 26 |
26 | 36 | 3328 |
26 | 48 | 104 |
26 | 52 | 52 |
27 | 0 | 512 |
27 | 4 | 13824 |
28 | 0 | 16900 |
28 | 4 | 240240 |
28 | 8 | 296688 |
28 | 16 | 94136 |
28 | 20 | 3696 |
28 | 28 | 7280 |
28 | 32 | 10892 |
28 | 40 | 2688 |
28 | 52 | 336 |
28 | 56 | 112 |
28 | 64 | 56 |
29 | 0 | 2 |
29 | 4 | 58 |
30 | 0 | 146854 |
30 | 4 | 2856780 |
30 | 8 | 3657600 |
30 | 12 | 1151400 |
30 | 16 | 2268360 |
30 | 20 | 528600 |
30 | 24 | 675840 |
30 | 28 | 240480 |
30 | 32 | 447480 |
30 | 36 | 40980 |
30 | 40 | 92160 |
30 | 44 | 72000 |
30 | 48 | 38460 |
30 | 52 | 1080 |
30 | 56 | 28800 |
30 | 60 | 5160 |
30 | 64 | 7410 |
30 | 68 | 120 |
30 | 72 | 1920 |
30 | 76 | 1320 |
30 | 80 | 300 |
30 | 92 | 120 |
30 | 96 | 60 |