query: There does not exist a polynomial $p(x)$ with integer coefficients which gives a prime number $\forall x\in \mathbb{Z}$
positive_passages.docid: 21862
positive_passages.text: Proof of lack of pure prime producing polynomials.
negative_passages.docid: 69523
negative_passages.text: Prove that polynomial of degree $4$ with real roots cannot have $\pm 1$ as coefficients (IITJEE)
negative_passages.docid: 70523
negative_passages.text: Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.
negative_passages.docid: 58927
negative_passages.text: Integer coefficient polynomial $p(x)$ has no integer roots of $\,p(0)$ and $p(1)$ are odd [Parity Root Test]
negative_passages.docid: 87101
negative_passages.text: Irreducible but not prime in $K[X,Y]/(X^2-Y^3)$
negative_passages.docid: 99000
negative_passages.text: No polynomial with integer coefficients satisfies the conditions
negative_passages.docid: 37315
negative_passages.text: not both $2^n-1,2^n+1$ can be prime.
negative_passages.docid: 106914
negative_passages.text: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree
negative_passages.docid: 111967
negative_passages.text: Let $p$ be a prime number. Prove that there exists $x∈Z$ for which $p|x^2−x+ 3$ if and only if there exists $y∈Z$ for which $p|y^2−y+ 25$.
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 42932
negative_passages.text: Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
negative_passages.docid: 27540
negative_passages.text: A polynomial can not have infinite number of constant values
negative_passages.docid: 105679
negative_passages.text: Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients?
negative_passages.docid: 98731
negative_passages.text: Show that there is no epimorphism from ($\mathbb{Q}$, +) to ($\mathbb{Z}$, +)
negative_passages.docid: 120940
negative_passages.text: An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$
negative_passages.docid: 63493
negative_passages.text: Why 4 is not a primitive root modulo p for any prime p?
negative_passages.docid: 71788
negative_passages.text: Prove existence of multiplicative inverse modulo p when p is prime