query: Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$
positive_passages.docid: 15753
positive_passages.text: Solve the following equation in positive integers $x$ and $y$
negative_passages.docid: 47809
negative_passages.text: If $x,y\in\mathbb Z$, solve $x^2+xy+y^2=x^2y^2$.
negative_passages.docid: 86285
negative_passages.text: Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
negative_passages.docid: 38849
negative_passages.text: Solve $x^y \, = \, y^x$
negative_passages.docid: 146753
negative_passages.text: Find $x$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$
negative_passages.docid: 88333
negative_passages.text: Find all integral solutions of $y^2=x^3+7$
negative_passages.docid: 22112
negative_passages.text: Find all integer solutions to $x^2+4=y^3$.
negative_passages.docid: 114489
negative_passages.text: Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
negative_passages.docid: 46668
negative_passages.text: How to find integer solutions to $M^2=5N^2+2N+1$?
negative_passages.docid: 23866
negative_passages.text: Do there exist an infinite number of integer-solutions $(x,y,z)$ of $x^x\cdot y^y=z^z$ where $1\lt x\le y$?
negative_passages.docid: 12967
negative_passages.text: Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
negative_passages.docid: 148906
negative_passages.text: Finding integers $m,n$ such that $(m-n)^2=4mn/(m+n-1)$
negative_passages.docid: 20486
negative_passages.text: Numbers representable as $x^2 + 2y^2$
negative_passages.docid: 124223
negative_passages.text: Find all pairs (x, y) of real numbers such that $16^{x^2+y} + 16^{x+y^2}=1$
negative_passages.docid: 20491
negative_passages.text: How to solve this equation $x^{2}=2^{x}$?
negative_passages.docid: 3390
negative_passages.text: finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$
negative_passages.docid: 35504
negative_passages.text: The Diophantine equation $x^2 + 2 = y^3$