Timeline for cosine of rational multiples of Pi take values of equal difference
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 21, 2017 at 13:36 | vote | accept | Andrew | ||
| Apr 19, 2017 at 17:24 | vote | accept | Andrew | ||
| Apr 19, 2017 at 18:15 | |||||
| Apr 19, 2017 at 16:38 | answer | added | Andrew | timeline score: 0 | |
| Apr 15, 2017 at 19:43 | answer | added | Anthony Quas | timeline score: 2 | |
| Apr 15, 2017 at 19:05 | comment | added | Andrew | @AnthonyQuas I have problem to see "If your equation is satisfied, it gives rise to an equation in $\zeta$ with rational coefficients of degree at most 5N/6 satisfied by $\zeta$." What is the primitive Nth root of unity in the current problem? Maybe you mean converting the cosines into the Euler complex form? But then where does the 5N/6 come from? Also the Euler formula would give several different roots of unity, not just a single $\zeta$. Could you please elaborate further? | |
| Apr 15, 2017 at 15:30 | comment | added | Anthony Quas | If $\zeta$ is a primitive $N$th root of unity, its minimal polynomial over the rationals is the $N$th cyclotomic polynomial, $\Phi_N$. This of degree $\phi(N)$. Any rational polynomial that has $\zeta$ as a root is a multiple of $\Phi_N$. If your equation is satisfied, it gives rise to an equation in $\zeta$ with rational coefficients of degree at most $5N/6$ satisfied by $\zeta$. This is a contradiction if $\phi(N)>5N/6$. | |
| Apr 15, 2017 at 13:52 | comment | added | Andrew | @AnthonyQuas After some learning in the Euler phi-function, I cannot see how it is related to the current problem in the way Anthony proposed, could you please elaborate? Thank you very much! | |
| Apr 15, 2017 at 1:57 | review | Close votes | |||
| Apr 15, 2017 at 8:29 | |||||
| Apr 15, 2017 at 1:39 | comment | added | user6976 | @FedorPetrov: You are right! | |
| Apr 14, 2017 at 23:26 | comment | added | Anthony Quas | This is certainly true if $N$ is prime, or more generally if $\phi(N)>\frac 56N$, where $\phi$ is the Euler phi-function, since a relation of the type you're interested in gives rise to a polynomial with integer coefficients satisfied by $e^{2\pi i/N}$ of degree at most $5N/6$. | |
| Apr 14, 2017 at 19:11 | comment | added | Fedor Petrov | Yes, but OP asks about the case when $m_1-m_2\ne m_2-m_3$ | |
| Apr 14, 2017 at 19:05 | comment | added | Fedor Petrov | @MarkSapir so what? | |
| Apr 14, 2017 at 18:35 | history | edited | Andrew | CC BY-SA 3.0 |
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| Apr 14, 2017 at 18:27 | review | First posts | |||
| Apr 14, 2017 at 18:55 | |||||
| Apr 14, 2017 at 18:26 | history | asked | Andrew | CC BY-SA 3.0 |