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I have a question concerning cyclotomic polynomials valuated at primes. I first state it in the easiest possible form.

There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that, if $p$ is a prime, then either $p^2+p+1$ is divisible by a prime larger than $c$, or $p\le f(c)$.

I have no idea in how to prove this, or whether it is already known. The more general form is the following.

Let $n\ge 3$ be an integer and let $\Phi_n(x)$ be the $n$th cyclotomic polynomial. There exists a function such that, if $p$ is prime, then either $\Phi_n(p)$ is divisible by a prime larger than $c$, or $p\le f(c)$.

When $n=2$ one cannot hope for such a result. But whenever $n>2$, I have no counterexamples.

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    $\begingroup$ If I am interpreting the question correctly, what you are asking is that either $\Phi_n(p)$ is not $g(p)$-smooth for some function $g$ tending to infinity, and to give some sort of growth rate for $g$. I believe this is possible using some type of quantitative analogue of the $abc$-conjecture, for example due to Stewart and Yu. $\endgroup$ Commented Sep 22, 2023 at 16:51
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    $\begingroup$ What you really want to prove is this: for fixed $n$, the largest prime factor of $\Phi_n(p)$ tends to infinity (as $p$ runs through the primes). $\endgroup$ Commented Sep 22, 2023 at 18:16
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    $\begingroup$ Following up on GH from MO's comment, what you want follows from work of Siegel; a quantiative version could be deduced from this paper of Shorey and Tijdeman: numdam.org/item/CM_1976__33_2_187_0 $\endgroup$ Commented Sep 22, 2023 at 21:45
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    $\begingroup$ @so-calledfriendDon I suggest that you turn your comment into an answer so that this question can be closed. $\endgroup$ Commented Sep 23, 2023 at 18:25
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    $\begingroup$ @GHfromMO Done. $\endgroup$ Commented Sep 23, 2023 at 19:14

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Following up on the comment of GH from MO: You want that for each fixed $n\ge 3$, the largest prime factor of $\Phi_n(p)$ tends to infinity as $p$ tends to infinity (through prime values). Work of Siegel implies that the largest prime factor of $f(n)$ tends to infinity as $n$ tends to infinity (through positive integers) whenever $f(T) \in \mathbb{Z}[T]$ has at least two distinct roots. Since $\Phi_n(T)$ has $\phi(n)$ distinct roots, and $\phi(n) \ge 2$ whenever $n\ge 3$, Siegel's result does what you want. In fact, Siegel's theorem can be put in quantitative form: For polynomials $f$ satisfying the above conditions, the largest prime factor of $f(n)$ is $\gg_{f} \log\log{n}$, as $n\to\infty$. See:

On the greatest prime factors of polynomials at integer points. Shorey, T. N.; Tijdeman, R. Compositio Mathematica, Volume 33 (1976) no. 2, pp. 187-195.

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