query: Find all primes $p$ such that $p^3-4p+9=x^2$, where $x$ is a positive integer.
positive_passages.docid: 21898
positive_passages.text: Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square.
negative_passages.docid: 29811
negative_passages.text: Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
negative_passages.docid: 32618
negative_passages.text: Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
negative_passages.docid: 144928
negative_passages.text: Find all primes $p$ such that $\phi(x)=x^{13}$ is a homomorphism $\Bbb Z/p\Bbb Z \rightarrow \Bbb Z/p\Bbb Z$.
negative_passages.docid: 130846
negative_passages.text: Prove or disprove: If $a \in \mathbb{Z}$, $a > 0$, and $p$ is a prime such that $p^4 \mid a^3$, then $p^2 \mid a$.
negative_passages.docid: 65826
negative_passages.text: Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
negative_passages.docid: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 114489
negative_passages.text: Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
negative_passages.docid: 4943
negative_passages.text: If $p,q,r$ are all primes,and $p|qr-1$,$q|pr-1$ and $r|pq-1$,find all possible values of $pqr$.
negative_passages.docid: 46821
negative_passages.text: Finding roots in $\mathbb{Z}_p$
negative_passages.docid: 87282
negative_passages.text: Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$
negative_passages.docid: 10403
negative_passages.text: Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
negative_passages.docid: 52530
negative_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 146203
negative_passages.text: if p is prime then 1920 divides $p^4-10p^2+9$
negative_passages.docid: 80399
negative_passages.text: Is $29$ the only prime of the form $p^p+2$?
negative_passages.docid: 26461
negative_passages.text: Integer solutions of $x^3-x+9=5y^2$
negative_passages.docid: 95162
negative_passages.text: For a odd prime $p, \exists a,b$ such that $a^2 + ab + b^2 \equiv 0 \pmod{p} \iff \exists x,y $ such taht $x^2 + xy + y^2 = p$?