query: If $p, q$ are primes and $p$ does not divide $n-1$ and $p$ divides $n^q-1$. How to prove that $q|p-1$.
positive_passages.docid: 113763
positive_passages.text: Let $p$,$\ q$ be prime numbers and $n\in \mathbb{N}$ such that $p \nmid n-1$. If $p \mid n^q-1$, then $q \mid p-1$.
negative_passages.docid: 29572
negative_passages.text: Proof that $2^n-1$ does not always generate primes when primes are plugged in for $n$?
negative_passages.docid: 144622
negative_passages.text: Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
negative_passages.docid: 32618
negative_passages.text: Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
negative_passages.docid: 112631
negative_passages.text: Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$
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: 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: 5206
negative_passages.text: Highest power of a prime $p$ dividing $N!$
negative_passages.docid: 85042
negative_passages.text: Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$ is prime [Wolstenholme's theorem]
negative_passages.docid: 131551
negative_passages.text: $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes
negative_passages.docid: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 70463
negative_passages.text: Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
negative_passages.docid: 10861
negative_passages.text: Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$?
negative_passages.docid: 75733
negative_passages.text: Divisibility of prime numbers
negative_passages.docid: 26503
negative_passages.text: Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
negative_passages.docid: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number