Prove that there are infinitely many primes $p$ such that $x^{10} + x + 1 \equiv 0 \mod p$ has at least one solution $x\in\mathbb{Z}$.

Question

Prove that there are infinitely many primes $p$ such that

$$x^{10} + x + 1 \equiv 0 \mod p$$

has at least one solution $x\in\mathbb{Z}$.

I believe I should be doing a proof by contradiction but I cannot figure out where it arises. Any help will be appreciated! thank you!

Answer 1

Yes, a proof by contradiction is a good way to proceed. Assume that the property holds only for a finite set of primes $p_1,p_2,\dots, p_n$ then let $x:=p_1p_2\cdots p_n-1$. It follows that for $i=1,\dots,n$, $$N:=x^{10}+x+1\equiv 1-1+1=1 \pmod{p_i}.$$ Now consider $q$ be a prime which divides $N$. What may we conclude?