query: Which primes $p$ satisfy $n^2 \equiv -1 \mod p$ for a perfect square $n^2$?
positive_passages.docid: 153869
positive_passages.text: Under what $p$, $-1$ is a square in $\mathbf F_p$?
negative_passages.docid: 114346
negative_passages.text: Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$?
negative_passages.docid: 130499
negative_passages.text: How can $p^{q+1}+q^{p+1}$ be a perfect square?
negative_passages.docid: 75318
negative_passages.text: How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
negative_passages.docid: 75528
negative_passages.text: $x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
negative_passages.docid: 59782
negative_passages.text: Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?
negative_passages.docid: 78917
negative_passages.text: Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
negative_passages.docid: 18053
negative_passages.text: Find all prime $p$ such that $x^2=-1$ has a solution in $\mathbb{Z}/p\mathbb{Z}$
negative_passages.docid: 4129
negative_passages.text: Solvability of $a \equiv x^2 \mod b$
negative_passages.docid: 56328
negative_passages.text: Find p is the prime number which $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ both are square number.
negative_passages.docid: 24820
negative_passages.text: Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
negative_passages.docid: 40156
negative_passages.text: Determine all primes $p$ for which $5$ is a quadratic residue modulo $p$
negative_passages.docid: 102234
negative_passages.text: Show $a^p \equiv a \mod 2p$
negative_passages.docid: 118629
negative_passages.text: $p,q$ odd primes, $q\mid a^p-1\Rightarrow q\mid a-1$ or $\,2p\mid q-1$
negative_passages.docid: 119630
negative_passages.text: For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
negative_passages.docid: 8034
negative_passages.text: Can $n!$ be a perfect square when $n$ is an integer greater than $1$?