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My naive question 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.

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  • $\begingroup$ This resembles Theorem 2.4 (or thereabouts) in notes of Jameson that helped answer a question of mine (phi_n(p) is like p^(phi(n))). When I remember the link I will post it. However Jameson's result is about numbers, not polynomials. Gerhard "Should Text Less, Drive More" Paseman, 2018.01.15. $\endgroup$ – Gerhard Paseman Jan 16 '18 at 2:18
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This note (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.

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  • $\begingroup$ Thank you! Indeed, the result is true and a stronger version of it is in Curtis/Reiner. $\endgroup$ – Mark Sapir Jan 16 '18 at 3:14

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