query: Show that only one prime can be expressed as $n^3-1$ for some positive integer $n$
positive_passages.docid: 10364
positive_passages.text: Is $7$ the only prime followed by a cube?
negative_passages.docid: 51382
negative_passages.text: Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
negative_passages.docid: 34954
negative_passages.text: If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
negative_passages.docid: 75685
negative_passages.text: Prove that there is prime number and natural so $n\times q$ digits are $1$.
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 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: 144447
negative_passages.text: Show that $504 \mid (n^9 − n^3 )$ for any integer $n$
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: 102889
negative_passages.text: If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime.
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 126367
negative_passages.text: Let $p$, $q$ be prime numbers such that $n^{3pq} – n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$.
negative_passages.docid: 61546
negative_passages.text: Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
negative_passages.docid: 74512
negative_passages.text: How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
negative_passages.docid: 51520
negative_passages.text: Prove that the only prime triple is 3, 5, 7
negative_passages.docid: 111037
negative_passages.text: Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
negative_passages.docid: 45144
negative_passages.text: Another way to write least prime factor.
negative_passages.docid: 16621
negative_passages.text: Help understand the proof of infinitely many primes of the form $4n+3$