query: Prove that $4((p-1)!+1)\equiv -p\mod{(p(p+2)})$
positive_passages.docid: 45341
positive_passages.text: $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$
negative_passages.docid: 5708
negative_passages.text: Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 66861
negative_passages.text: Show that $11^{10} \equiv 1 (\mod 100)$
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 71788
negative_passages.text: Prove existence of multiplicative inverse modulo p when p is prime
negative_passages.docid: 32762
negative_passages.text: Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
negative_passages.docid: 21250
negative_passages.text: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
negative_passages.docid: 75528
negative_passages.text: $x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
negative_passages.docid: 21629
negative_passages.text: Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
negative_passages.docid: 151961
negative_passages.text: Irreducibility of $x^{p(p-1)}+x^{p(p-2)}+\cdots+x^p+1$ over $\mathbb Q$
negative_passages.docid: 17601
negative_passages.text: How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
negative_passages.docid: 116686
negative_passages.text: $p=2^n+1$. Prove that every quadratic nonresidue modulo $p$ is a primitive root modulo $p$
negative_passages.docid: 38805
negative_passages.text: Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 124995
negative_passages.text: $p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.