query: Find all primes $p$ such that $x^3+x+1\equiv0\pmod p$ has $3$ incongruent solutions.
positive_passages.docid: 21874
positive_passages.text: Solve $x^3+x \equiv 1 \pmod p$
negative_passages.docid: 9039
negative_passages.text: What is the best way to solve modular arithmetic equations such as $9x \equiv 33 \pmod{43}$?
negative_passages.docid: 1702
negative_passages.text: Proving that there are infinitely many primes with remainder of 2 when divided by 3
negative_passages.docid: 94017
negative_passages.text: If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
negative_passages.docid: 69858
negative_passages.text: Congruence Equation $3n^3+12n^2+13n+2\equiv0,\pmod{2\times3\times5}$
negative_passages.docid: 133925
negative_passages.text: Find All Solutions to System of Congruence
negative_passages.docid: 75528
negative_passages.text: $x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
negative_passages.docid: 94917
negative_passages.text: Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
negative_passages.docid: 146512
negative_passages.text: Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
negative_passages.docid: 110809
negative_passages.text: What can primes, except 2, 3, and 5, be congruent to $\pmod {30}$?
negative_passages.docid: 18634
negative_passages.text: Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
negative_passages.docid: 107990
negative_passages.text: Does there exist an integer $x$ satisfying the following congruences?
negative_passages.docid: 107381
negative_passages.text: Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$
negative_passages.docid: 66851
negative_passages.text: If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
negative_passages.docid: 102204
negative_passages.text: Factoring product of two primes from solutions of congruence
negative_passages.docid: 89590
negative_passages.text: The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$