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Number theory Difficulty 5.3 AIME, harder Prove it

Let P(x)=x4−x3−3x2−x+1P(x)=x^{4}-x^{3}-3 x^{2}-x+1. Show that there are infinitely many integers nn such that P(3n)P\left(3^{n}\right) is not prime.

Solution

We observe that 32≡−1( mod 5)3^{2} \equiv-1(\bmod 5) and 34≡1( mod 5)3^{4} \equiv 1(\bmod 5). Let n⩾1n \geqslant 1. Let x=34n+1x=3^{4 n+1}, then x=(34)n×3≡3( mod 5)x=\left(3^{4}\right)^{n} \times 3 \equiv 3(\bmod 5), so P(x)≡34−33−33−3+1≡1+3+3−3+1≡0P(x) \equiv 3^{4}-3^{3}-3^{3}-3+1 \equiv 1+3+3-3+1 \equiv 0 ( mod 5)(\bmod 5).
On the other hand, P(x)>x4−x3−3x3−x3=x3(x−5)>x−5>34n−5>5P(x)>x^{4}-x^{3}-3 x^{3}-x^{3}=x^{3}(x-5)>x-5>3^{4 n}-5>5, so P(34n+1)P\left(3^{4 n+1}\right) is not prime.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.