Prove that there are infinitely many primes p, such that the congruence $x^7$ + $3x^5 + 1 \equiv 0 \mod p$ has a solution

Question

How would you go about solving this? What kind of proof should i use? Any help will be much appreciated, thank you!

Instead, you may also use something called Chebotarev density theorem (which proves a stronger result, that this kind of prime numbers have positive density). — WhatsUp, Jan 27 '20 at 12:20

score 0 · Accepted Answer · answered Jan 28 '20 at 00:40

Since $1+3+1=5$, there is at least one such prime. So, suppose that there are only a finite number of such primes and let $P$ be their product.

Let $q$ be any prime dividing $P^7+3P^5+1$. Then $q$ is not a factor of $P$, a contradiction.

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