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% |

31 | 1180 | 2147483648 | 0.00005% |

32 | 2228224 | 4294967296 | 0.05187% |

33 | 87788 | 8589934592 | 0.00102% |

34 | 2359468 | 17179869184 | 0.01373% |

35 | 17098 | 34359738368 | 0.00004% |

36 | 52492960 | 68719476736 | 0.07638% |

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 |

31 | 0 | 2 |

31 | 4 | 62 |

31 | 20 | 620 |

31 | 32 | 496 |

32 | 0 | 65536 |

32 | 4 | 1048576 |

32 | 8 | 524288 |

32 | 12 | 524288 |

32 | 28 | 65536 |

33 | 0 | 2054 |

33 | 4 | 67848 |

33 | 12 | 13068 |

33 | 16 | 66 |

33 | 20 | 4224 |

33 | 36 | 264 |

33 | 44 | 132 |

33 | 48 | 132 |

34 | 0 | 131074 |

34 | 4 | 2228292 |

34 | 16 | 34 |

34 | 68 | 68 |

35 | 0 | 228 |

35 | 4 | 5600 |

35 | 8 | 5320 |

35 | 12 | 3080 |

35 | 16 | 1190 |

35 | 20 | 140 |

35 | 24 | 280 |

35 | 28 | 140 |

35 | 32 | 140 |

35 | 36 | 420 |

35 | 40 | 280 |

35 | 44 | 140 |

35 | 72 | 140 |

36 | 0 | 1000000 |

36 | 4 | 21600000 |

36 | 8 | 12960000 |

36 | 12 | 7200000 |

36 | 16 | 3600000 |

36 | 20 | 4320000 |

36 | 24 | 1440000 |

36 | 32 | 360000 |

36 | 68 | 12960 |

Actually, this problem has some variations, for example:

Consider $x_i = 1, 0$ or $x_i = \pm 1,0$ instead of $x_i = \pm 1$.

Consider remove the constraint $x_i^2=1$.

Consider $x_i \in \mathbb{Q}$ or $x_i \in \mathbb{R}$ instead of $x_i \in \mathbb{Z}$.

Any comment/answer to this problem and its variations will be appreciated.

Here are my motivations:

First, in algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity. From this equation, maybe we can find out more interesting formulas.

Second, in linear algebra, the order of Hadamard matrices is usually $4k$, where $k=1,2,3,...$, and here $\lvert z \rvert^2$ is usually $4k$ too. Although Hadamard conjecture is still an open problem, maybe one day this conjecture will be proven and we will find out that it is not a coincidence....

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