query: Show that if $p>5$ is prime, then $240 \mid (p^4-1)$
positive_passages.docid: 18405
positive_passages.text: What is the greatest integer that divides $p^4-1$ for every prime number $p$ greater than $5$?
negative_passages.docid: 143543
negative_passages.text: Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
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: 118574
negative_passages.text: prove that $24 \mid a(a^2-1)$
negative_passages.docid: 117516
negative_passages.text: Prime trios of the form $p$,$p+2$,$p+4$
negative_passages.docid: 63702
negative_passages.text: There is a prime between $n$ and $n^2$, without Bertrand
negative_passages.docid: 43567
negative_passages.text: Some hints for "If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$"
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 12365
negative_passages.text: Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
negative_passages.docid: 77430
negative_passages.text: Prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$
negative_passages.docid: 78917
negative_passages.text: Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
negative_passages.docid: 65826
negative_passages.text: Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
negative_passages.docid: 105442
negative_passages.text: Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
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: 116363
negative_passages.text: If $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$ , then prove that $P(x)$ is divisible by $x-1$