query: Let $p$ and $q$ be distinct primes. $\frac{1}{x}+\frac{1}{y}=\frac{1}{pq}$
positive_passages.docid: 157264
positive_passages.text: Solve $\frac1x+\frac1y=\frac1{pq}$
negative_passages.docid: 144814
negative_passages.text: If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?
negative_passages.docid: 140599
negative_passages.text: Inequality Proof in Real Numbers: If $p+q=1$ then $pq\le\frac14$
negative_passages.docid: 21846
negative_passages.text: Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions?
negative_passages.docid: 156646
negative_passages.text: If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$
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: 25835
negative_passages.text: Structure of groups of order $pq$, where $p,q$ are distinct primes.
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: 20980
negative_passages.text: Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$
negative_passages.docid: 58182
negative_passages.text: How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
negative_passages.docid: 158863
negative_passages.text: How to prove $\sum_p p^{-2} < \frac{1}{2}$?
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: 155886
negative_passages.text: let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.
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: 50413
negative_passages.text: Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
negative_passages.docid: 158161
negative_passages.text: $\mathbb{Q}(\sqrt p_{1}+\sqrt p_{2})=\mathbb{Q}(\sqrt p_{1},\sqrt p_{2})$ for $p_{1},p_{2}$ primes.