query: Find all solutions of $a^b = b^a$
positive_passages.docid: 18761
positive_passages.text: Given that $x^y=y^x$, what could $x$ and $y$ be?
negative_passages.docid: 80982
negative_passages.text: Solve $a^3+b^3+3ab=1$ with $(a,b)\in \Bbb{Z}^2$
negative_passages.docid: 69328
negative_passages.text: (USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$
negative_passages.docid: 38379
negative_passages.text: Showing that $e^z=z$ has infinitely many solutions
negative_passages.docid: 36602
negative_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$
negative_passages.docid: 33542
negative_passages.text: When is $A- B = B- A$?
negative_passages.docid: 94037
negative_passages.text: Solving the matrix equation $AP=PB$, where $A$ and $B$ are similar
negative_passages.docid: 18203
negative_passages.text: Diophantine equation $a^2+b^2=c^2+d^2$
negative_passages.docid: 4129
negative_passages.text: Solvability of $a \equiv x^2 \mod b$
negative_passages.docid: 148983
negative_passages.text: If $a$, $b$, $c$ are the roots of $x^3-6x^2+3x+1=0$, find all possible values of $a^2b+b^2c+c^2a$
negative_passages.docid: 147837
negative_passages.text: Find the parameter values satisfying conditions for all roots of an equation
negative_passages.docid: 24499
negative_passages.text: Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$
negative_passages.docid: 113458
negative_passages.text: How to factor $a^n - b^n$?
negative_passages.docid: 41977
negative_passages.text: If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
negative_passages.docid: 80817
negative_passages.text: Solving equations of the form $ae^x + bx +c =0$
negative_passages.docid: 69874
negative_passages.text: Are there any integer solutions to $a^3=b^2$?
negative_passages.docid: 66256
negative_passages.text: Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$