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The Question

How many unique values of $\cos(\frac{a\pi}{N})\cos(\frac{b\pi}{N})$ are there for the positive integers $a,b < N$ for a given $N$? I would like a function $f(N)$ which gives that number of unique values.

Upper Bound

It can be shown that for even $N$, $f(N)$ is inclusive upper bounded by $$f'(N) = (\sum_{n=1}^{N/2}n) + 1$$

In fact, for many values of N, $f(N) = f'(N)$. This upper bound is easier to show from the equivalent formulation of the problem as discussed at the bottom of this question.

Example

N = 12 is the smallest even N such that $f(N)\lt f'(N)$, there is one duplicate value for N = 12:

$$\cos(\frac{3\pi}{12})\cos(\frac{3\pi}{12}) = \cos(\frac{0\pi}{12})\cos(\frac{4\pi}{12})$$

So $f(12) = 21$, while $f'(12) = 22$.

Sequence

I have calculated the sequence $f(N)$ for $0<N<70$ as follows:

1, 2, 4, 4, 7, 7, 11, 11, 16, 16, 22, 21, 29, 29, 36, 37, 46, 45, 56, 56, 67, 67, 79, 77, 92, 92, 106, 106, 121, 116, 137, 137, 154, 154, 172, 170, 191, 191, 211, 211, 232, 232, 254, 254, 276, 277, 301, 299, 326, 326, 352, 352, 379, 377, 407, 407, 436, 436, 466, 458, 497, 497, 529, 529, 562, 560, 596, 596, 631

and for even integers $0 < N < 70$, the sequence of $f'(N) - f(N)$ is:

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 8, 0, 0, 2, 0

Neither of these integer sequences are present on https://oeis.org/.

Equivalent formulation

Where $\omega_N$ is the $N^{th}$ root of unity, how many unique values of $(\omega^a_N + \omega^{-a}_N)(\omega^b_N + \omega^{-b}_N)$ are there for a given N?

This question has arisen for me while analyzing the spectra of the tensor product of two same-sized cyclic groups. I can expand on this context if it would be helpful for solving the problem.

I do have some additional, various, disconnected insight into the problem. But for the sake of clarity of the question, I will end the question here. Thank you, any insight and/or discussion is appreciated.

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    $\begingroup$ In my paper, Rational products of sines of rational angles, Aequationes mathematicae volume 45, pages70–82(1993), I found all solutions of $\sin x\sin y=\sin z\sin w$ with $x,y,z,w$ rational multiples of $\pi$. There are only finitely many, with some mild restrictions on the variables. $\endgroup$ Commented Aug 5, 2020 at 6:19
  • $\begingroup$ Thanks Gerry, that sounds great. Where can I obtain a copy of your paper? $\endgroup$
    – Elliot E
    Commented Aug 5, 2020 at 6:24
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    $\begingroup$ A math library. Well, Springer Link website has it for sale, but I don't think it's worth what they want for it. oeis.org/A134869 looks a lot like every other term of your sequence. $\endgroup$ Commented Aug 5, 2020 at 6:27
  • $\begingroup$ That sequence is the same as $f'(x)$. It is crucially missing the information present in $f'(x) - f(x)$. $\endgroup$
    – Elliot E
    Commented Aug 5, 2020 at 6:34
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    $\begingroup$ I have found the following available text: annalesm.elte.hu/annales12-1969/Annales_1969_T-XII.pdf#page=147 on the equation $\cos\alpha_1 + \cos\alpha_2 + \cos\alpha_3 + \cos\alpha_4 = 0$. I am fairly sure this is an equivalent equation to my problem. I will attempt to relate them and post an answer to this question later if I am able to. $\endgroup$
    – Elliot E
    Commented Aug 5, 2020 at 19:48

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