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.

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.