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Let f(x) be a nonconstant polynomial with integer coefficients. Prove that there is some integer n such that |f(n)| is composite.

I'm just a little confused on how to begin. I am fairly certain we start by assuming that f(x) outputs only primes for all integers x, but I don't know where to go from there. Any help would be greatly appreciated.

Thanks!

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marked as duplicate by Martin R, Larry, Vinyl_cape_jawa, José Carlos Santos, Arnaud D. Oct 14 at 15:22

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Since you are only looking for a strategy to start with, not a solution:

$p:=f(0)$ and by assumption $p$ is prime. Now look at the values of $f(k\cdot p)$ and see if you can proove they are not prime.

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  • $\begingroup$ Is that different from math.stackexchange.com/a/193804/42969 (the answer to the first duplicate target)? $\endgroup$ – Martin R Oct 14 at 7:41
  • $\begingroup$ @MartinR It is the same solution, but since OP only asked for an idea, not a complete proof, I figured this would be more what he needs. $\endgroup$ – Kaligule Oct 14 at 10:08

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