query: Number of positive integer solutions of $\frac{1}{x}+\frac{1}{y}=\frac{1}{pq}$ for distinct primes $p$ and $q$
positive_passages.docid: 157264
positive_passages.text: Solve $\frac1x+\frac1y=\frac1{pq}$
negative_passages.docid: 107990
negative_passages.text: Does there exist an integer $x$ satisfying the following congruences?
negative_passages.docid: 26478
negative_passages.text: Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
negative_passages.docid: 114379
negative_passages.text: Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
negative_passages.docid: 134352
negative_passages.text: The number of possible factorizations of a positive integer.
negative_passages.docid: 135892
negative_passages.text: Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes
negative_passages.docid: 63893
negative_passages.text: Prime dividing the binomial coefficients
negative_passages.docid: 146491
negative_passages.text: Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$
negative_passages.docid: 46028
negative_passages.text: Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.
negative_passages.docid: 103832
negative_passages.text: Number of solutions of $x_1 + x_2 + x_3 + x_4 = 14$ such that $x_i \le 6$
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: 72857
negative_passages.text: Find the number of integral solutions of $2x + y + z = 20$?
negative_passages.docid: 44744
negative_passages.text: Integer solutions of $xy+9(x+y)=2006$
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: 95494
negative_passages.text: Number of solutions of an arithmetic function's equation
negative_passages.docid: 133101
negative_passages.text: Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial
negative_passages.docid: 44250
negative_passages.text: rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$