query: Prove that there are infinitely many primes $p$ such that $x^{10} + x + 1 \equiv 0 \mod p$ has at least one solution $x\in\mathbb{Z}$.
positive_passages.docid: 75528
positive_passages.text: $x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
negative_passages.docid: 22023
negative_passages.text: Infinitely number of primes in the form $4n+1$ proof
negative_passages.docid: 136279
negative_passages.text: Infinitely many primes $\equiv 3 \mod 4$
negative_passages.docid: 107381
negative_passages.text: Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$
negative_passages.docid: 54286
negative_passages.text: Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$
negative_passages.docid: 127945
negative_passages.text: Number of solutions to the congruence $x^q \equiv 1 \mod p$.
negative_passages.docid: 47162
negative_passages.text: Proving there are infinitely many primes of the form $a2^k+1.$
negative_passages.docid: 138409
negative_passages.text: Proof of infinitely many primes, clarification
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 132132
negative_passages.text: Are there infinitely many primes of the form $6^{2n}+1$ or only finitely many?
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 128107
negative_passages.text: find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
negative_passages.docid: 65000
negative_passages.text: prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$
negative_passages.docid: 73515
negative_passages.text: Primes Number Theory
negative_passages.docid: 68328
negative_passages.text: Infinitely many $n$ such that $2^n-n$ is divisible by a prime
negative_passages.docid: 59261
negative_passages.text: If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$.