query: Why are there $3$ solutions to $x^a = a^x$?
positive_passages.docid: 25078
positive_passages.text: What is the solution set for the equation$ x ^ y = y ^ x$ in positive real numbers, x not equal to y?
negative_passages.docid: 103223
negative_passages.text: If $x^a$ is $x$ multiplied $a$ times, then how does $x^{-1}$ make sense?
negative_passages.docid: 62850
negative_passages.text: Linear Algebra: If $A^3 = I$, does $A$ have to be $I$?
negative_passages.docid: 82137
negative_passages.text: Given a, b How many solutions exists for x, such that: $a \bmod{x}=b $
negative_passages.docid: 117352
negative_passages.text: Show $x^n+ax+b=0$ has most two solutions
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: 73079
negative_passages.text: Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$
negative_passages.docid: 15314
negative_passages.text: Number of solutions to $x^n \equiv 1 \mod p$
negative_passages.docid: 114185
negative_passages.text: Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$
negative_passages.docid: 51153
negative_passages.text: Number of solutions to $x^2\equiv a\pmod m$
negative_passages.docid: 38379
negative_passages.text: Showing that $e^z=z$ has infinitely many solutions
negative_passages.docid: 82571
negative_passages.text: Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
negative_passages.docid: 38636
negative_passages.text: A basic question about exponentiation: when is $a^{xy}=(a^x)^y$?
negative_passages.docid: 77595
negative_passages.text: Number of solutions of $x_1 + 2x_2 + 3x_3 + \cdots + n x_n = p$
negative_passages.docid: 114379
negative_passages.text: Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
negative_passages.docid: 106308
negative_passages.text: how do you solve $x^x=x$
negative_passages.docid: 112390
negative_passages.text: Solutions of $x^2 \equiv 1 \pmod N$