My naive [question](https://mathoverflow.net/questions/290822/cyclotomic-polynomials) may actually lead to something interesting. 


Let $\Phi\_m(x)$, $\Phi\_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $s(x), t(x)$ with rational coefficients such that $s\Phi\_m+t\Phi\_n=1$. Let $d$ be the least common denominator of the coefficients of the polynomials $s,t$ (of minimal degrees, $s,t$ obtained by the Euclidean algorithm). A Maple experiment shows that the following conjectures may be true. 


**Conjecture 1.** If the number $d$ is not equal to $1$ then $m$ divides $n$ and $n/m$ is a prime power.


**Conjecture 2.** If $n=mp^k$ where $p$ is prime, then $d=p$. 


**Question.** Are these conjectures true?


**Motivation:** $\Phi\_m$ and $\Phi\_n$ are co-prime in the ring of integers modulo any number that is co-prime with $d$. In particular, if $d=1$, then the reductions of $\Phi\_m$ and $\Phi\_n$ modulo any number would be relatively prime. 



Cyclotomic polynomials 2
[This note](https://projecteuclid.org/euclid.rmjm/1181071615) (see theorem 2.2) attributes this result to



> 
> Diederichsen, Fritz-Erdmann,
>  "Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz", Abh. Math. Sem. Hansischen Univ. 13, (1940). 357–412
> 
> 
> 


but also points out to pp. 550-554 in Curtis/Reiner "Methods of representation theory, with applications to finite groups and orders" vol.1.


