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Does there exist an irreducible non-linear polynomial $P(x)\in\mathbb{Z}[x]$ such that for any prime number $q$ there exists $t\in\mathbb{N}$ such that $q|P(t)$ ?

Also (dis)proving whether there exists such polynomial that satisfies the condition for all but finitely many prime numbers is of interest.

All I have reached at the moment has been the polynomial $P(x)=(x^2-3)(x^2-5)(x^2-15)$ which is clearly not irreducible but has no integer roots and satisfies the condition of the problem and I couldn't do much more.

Aryan
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    Is this already answered at https://math.stackexchange.com/questions/608919/is-it-true-that-if-fx-has-a-linear-factor-over-mathbbf-p-for-every-prim ? – Gerry Myerson May 23 '21 at 08:01

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As you suspected, $P$ must be linear, already for all but finitely many primes.

See: https://mathoverflow.net/questions/150810/irreducible-polynomials-with-a-root-modulo-almost-all-primes

GreginGre
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