query: Prove that every prime number greater than 3 is either one more or one less than a multiple of $6$
positive_passages.docid: 2425
positive_passages.text: Is that true that all the prime numbers are of the form $6m \pm 1$?
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: 66851
negative_passages.text: If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 75206
negative_passages.text: If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
negative_passages.docid: 32803
negative_passages.text: Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
negative_passages.docid: 48484
negative_passages.text: Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 71880
negative_passages.text: Prove that every prime factor of $12x^2+1$ has the form $6n+1$
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
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: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 51382
negative_passages.text: Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
negative_passages.docid: 134963
negative_passages.text: Proving a non-prime (composite) number $> 1$ has a divisor $d \leq\sqrt q$
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.