query: Under what $p$, $-1$ is a square in $\mathbf F_p$?
positive_passages.docid: 27287
positive_passages.text: $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
negative_passages.docid: 91935
negative_passages.text: If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
negative_passages.docid: 50596
negative_passages.text: $p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
negative_passages.docid: 134348
negative_passages.text: When is $(p - 1)! + 1$ a power of $p$?
negative_passages.docid: 86105
negative_passages.text: Can $\{(f(t),g(t)) \mid t\in [a,b]\}$ cover the entire square $[0,1] \times [0,1]$ ?
negative_passages.docid: 19202
negative_passages.text: $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 122741
negative_passages.text: $x^2$ is congruent to $-1 \bmod p$
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
negative_passages.docid: 121395
negative_passages.text: Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
negative_passages.docid: 40209
negative_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 8494
negative_passages.text: Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.
negative_passages.docid: 46746
negative_passages.text: How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
negative_passages.docid: 131108
negative_passages.text: Can $3p^4-3p^2+1$ be square number?
negative_passages.docid: 20526
negative_passages.text: Show that $PF.PG=b^2$ in a hyperbola
negative_passages.docid: 24107
negative_passages.text: Bijection $f:\mathcal{P}(A)\to(A\to \{0,1\})$
negative_passages.docid: 121957
negative_passages.text: Is $p\mapsto \|f\|_p$ continuous?