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3 clarity improvement

Dear MO_World,

I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning cyclotomic polynomials that I'm unable to tackle.

The question is this:

Let $p$ be a prime and let $p|n$. When is it the case that $\Phi_n(e^{2\pi i/p})=\pm e^{2\pi ij/p}$ for some $j$?

Here $\Phi_n$ denotes the $n$th cyclotomic polynomial.

I've experimented with Mathematica and have found there are non-trivial cases in which the condition holds, whereas for most cases it does not seem to hold.

Letting $c(n,p)=\Phi_n(e^{2\pi i/p})$, we have $c(105,3)=1$, but $c(105,5)$ and $c(105,7)$ are not on the unit circle. None of $c(15,3)$, $c(21,3)$, $c(15,5)$, $c(35,5)$, $c(21,7)$, $c(35,7)$ are on the unit circle; $c(40,2)=1$, but $c(50,2)=5$...

Not surprisingly it seems to be easiest for the condition to hold for small $p$ (p$. Also, using the relations$\Phi_{p^2n}(x)=\Phi_{pn}(x^p)$; and$\Phi_n(1)=q$if$n=q^k$for some prime$q$and an integer$k$, but$\Phi_n(1)=1$otherwise)otherwise, it's not hard to see that the condition holds whenever$p^2|n$, but$n$is not a power of$p$. Thanks for any more systematic suggestions... 2 added 444 characters in body Dear MO_World, I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning cyclotomic polynomials that I'm unable to tackle. The question is this: Let$p$be a prime and let$p|n$. When is it the case that$\Phi_n(e^{2\pi i/p})=\pm e^{2\pi ij/p}$for some$j$? Here$\Phi_n$denotes the$n$th cyclotomic polynomial. I've experimented with Mathematica and have found there are non-trivial cases in which the condition holds, whereas for most cases it does not seem to hold. Letting$c(n,p)=\Phi_n(e^{2\pi i/p})$, we have$c(105,3)=1$, but$c(105,5)$and$c(105,7)$are not on the unit circle. None of$c(15,3)$,$c(21,3)$,$c(15,5)$,$c(35,5)$,$c(21,7)$,$c(35,7)$are on the unit circle;$c(40,2)=1$, but$c(50,2)=5$. c(50,2)=5$...

Not surprisingly it seems to be easiest for the condition to hold for small $p$ (using the relations $\Phi_{p^2n}(x)=\Phi_{pn}(x^p)$; and $\Phi_n(1)=q$ if $n=q^k$ for some prime $q$ and an integer $k$, but $\Phi_n(1)=1$ otherwise), it's not hard to see that the condition holds whenever $p^2|n$, but $n$ is not a power of $p$.

Thanks for any more systematic suggestions...

1

# Cyclotomic polynomials evaluated at roots of unity

Dear MO_World,

I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning cyclotomic polynomials that I'm unable to tackle.

The question is this:

Let $p$ be a prime and let $p|n$. When is it the case that $\Phi_n(e^{2\pi i/p})=\pm e^{2\pi ij/p}$ for some $j$?

Here $\Phi_n$ denotes the $n$th cyclotomic polynomial.

I've experimented with Mathematica and have found there are non-trivial cases in which the condition holds, whereas for most cases it does not seem to hold.

Letting $c(n,p)=\Phi_n(e^{2\pi i/p})$, we have $c(105,3)=1$, but $c(105,5)$ and $c(105,7)$ are not on the unit circle. $c(40,2)=1$, but $c(50,2)=5$.

Thanks for any suggestions...