query: Proving $n^2 + n + 1$ has prime factors of the form 6m + 1 or 3.
positive_passages.docid: 17660
positive_passages.text: Show that the odd prime divisors of $n^2+n+1$ are of form $6k+1$ (exclude 3)
negative_passages.docid: 75011
negative_passages.text: Primes of the form $n^2+1$ - hard?
negative_passages.docid: 20881
negative_passages.text: Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
negative_passages.docid: 42885
negative_passages.text: Fermat primes relation to $2^n+1$
negative_passages.docid: 125188
negative_passages.text: how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
negative_passages.docid: 6918
negative_passages.text: Primes congruent to 1 mod 6
negative_passages.docid: 130842
negative_passages.text: Prime factors + number of Divisors
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 74512
negative_passages.text: How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 132132
negative_passages.text: Are there infinitely many primes of the form $6^{2n}+1$ or only finitely many?
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: 96894
negative_passages.text: Prove $f_{n+1}^2+f_n^2=f_{2n+1}$
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 34954
negative_passages.text: If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
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: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$