query: If $p$ is a prime, then $(p-1)! \equiv -1 (mod p)$
positive_passages.docid: 2218
positive_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 114346
negative_passages.text: Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$?
negative_passages.docid: 100984
negative_passages.text: Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
negative_passages.docid: 50413
negative_passages.text: Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
negative_passages.docid: 34692
negative_passages.text: Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 152591
negative_passages.text: For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?
negative_passages.docid: 66861
negative_passages.text: Show that $11^{10} \equiv 1 (\mod 100)$
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
negative_passages.docid: 123033
negative_passages.text: Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
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: 85287
negative_passages.text: Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$
negative_passages.docid: 110752
negative_passages.text: $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
negative_passages.docid: 118039
negative_passages.text: How prove $p!|(a+1)$ if $p!|a^p+1$
negative_passages.docid: 126725
negative_passages.text: If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
negative_passages.docid: 116585
negative_passages.text: $r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
negative_passages.docid: 158106
negative_passages.text: Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime