query: Number of ordered pairs of primes (p,q) such that pq divides (2^p+2^q).
positive_passages.docid: 137230
positive_passages.text: Find all pairs of primes $p,q$ such that $pq \mid 2^p +2^q$
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: 125330
negative_passages.text: How to find element g order q, given 2 large primes p and q where q|p-1
negative_passages.docid: 48502
negative_passages.text: number of ordered pairs of integers $(x,y)$ satisfying the equation
negative_passages.docid: 43591
negative_passages.text: How to factor numbers that are the product of two primes
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
negative_passages.docid: 64148
negative_passages.text: If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
negative_passages.docid: 42932
negative_passages.text: Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
negative_passages.docid: 115726
negative_passages.text: For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
negative_passages.docid: 120554
negative_passages.text: highest power of prime $p$ dividing $\binom{m+n}{n}$
negative_passages.docid: 47162
negative_passages.text: Proving there are infinitely many primes of the form $a2^k+1.$
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
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: 144821
negative_passages.text: Number of ordered pairs of subsets
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: 23885
negative_passages.text: Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
negative_passages.docid: 97021
negative_passages.text: Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?