Beedgighref28n

2022-04-21

How prove $\mathrm{cos}\left(\frac{2\pi }{17}\right)+\mathrm{cos}\left(\frac{18\pi }{17}\right)+\mathrm{cos}\left(\frac{26\pi }{17}\right)+\mathrm{cos}\left(\frac{30\pi }{17}\right)=\frac{\sqrt{17}-1}{4}$
Regards that value of $\mathrm{cos}\left(2\frac{\pi }{17}\right)$ I can't find the easy way to solve that expression.
Even if I had time, I wouldn't try that method to find the all roots others cosines expressions. IMHO

gonzakunti2

Let p be an odd ' number. Then
${g}_{p}=\sum _{k=0}^{p-1}\mathrm{exp}\left(2\pi i\frac{{k}^{2}}{p}\right)$
is a quadratic Gauss sum. Gauss proved that ${g}_{p}=\sqrt{p}$ or $i\sqrt{p}$ according to whether $p\equiv 1$ or . It is quite easy to prove this up to sign, but hard to prove the sign.
So ${g}_{17}=\sqrt{17}$. Therefore
$\sqrt{17}=1+2\mathrm{exp}\left(2\pi \frac{i}{17}\right)+2\mathrm{exp}\left(8\pi \frac{i}{17}\right)+2\mathrm{exp}\left(18\pi \frac{i}{17}\right)+2\mathrm{exp}\left(32\pi \frac{i}{17}\right)+2\mathrm{exp}\left(16\pi \frac{i}{17}\right)+2\mathrm{exp}\left(4\pi \frac{i}{17}\right)+2\mathrm{exp}\left(30\pi \frac{i}{17}\right)+2\mathrm{exp}\left(26\pi \frac{i}{17}\right)$
$=1+4\mathrm{cos}\left(2\frac{\pi }{17}\right)+4\mathrm{cos}\left(18\frac{\pi }{17}\right)+4\mathrm{cos}\left(26\frac{\pi }{17}\right)+4\mathrm{cos}\left(30\frac{\pi }{17}\right)$

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