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Let $P(x)$ be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers $n$ such that $P(n)$ is composite.

 Oct 29, 2019
 #1
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Let P(x) = x^2.  Then P(1) = 1, p(2) = 4, p(3) = 9, so p(n) is always a perfect square.  You can do this for any polynomial that factors, like P(x) = x^3 and P(x) = x(x + 1).

 Oct 29, 2019
 #2
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By the prime number theorem, the nth prime is like n*log n.  This is not a polynomial, so there must be an n such that p(n) is composite.

 Oct 30, 2019

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