query: Find all positive integers $x$ and $y$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{p}.$
positive_passages.docid: 21846
positive_passages.text: Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions?
negative_passages.docid: 95490
negative_passages.text: What are the positive integer solutions to $x^2-x+1 = y^3$?
negative_passages.docid: 77133
negative_passages.text: $g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.
negative_passages.docid: 140948
negative_passages.text: Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds.
negative_passages.docid: 29811
negative_passages.text: Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
negative_passages.docid: 21874
negative_passages.text: Solve $x^3+x \equiv 1 \pmod p$
negative_passages.docid: 77864
negative_passages.text: Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$
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: 106868
negative_passages.text: Prove that $x+\frac1{4x} ≥ 1$ for $x>0$
negative_passages.docid: 142724
negative_passages.text: Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
negative_passages.docid: 136586
negative_passages.text: Making $x+\frac1y$, $y+\frac1z$, and $z+\frac1x$ all integer.
negative_passages.docid: 146169
negative_passages.text: Integer solutions to $\frac{1}{a} + \frac{1}{b} = \frac{1}{12}$.
negative_passages.docid: 157746
negative_passages.text: Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$
negative_passages.docid: 15268
negative_passages.text: Find all $x,y \in \mathbb N^*$ such that $\frac{1}{x} + \frac{1}{y}=5$
negative_passages.docid: 70406
negative_passages.text: Find all values x, y and z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes.
negative_passages.docid: 155559
negative_passages.text: Quickest way to determine a polynomial with positive integer coefficients
negative_passages.docid: 123674
negative_passages.text: Show that $y = \frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive.