query: Prove rigorously: $x^2=2^x$ has exactly $3$ real solutions
positive_passages.docid: 20491
positive_passages.text: How to solve this equation $x^{2}=2^{x}$?
negative_passages.docid: 94700
negative_passages.text: Integer solutions for $x^3+2=y^2$?
negative_passages.docid: 7279
negative_passages.text: $|2^x-3^y|=1$ has only three natural pairs as solutions
negative_passages.docid: 111467
negative_passages.text: find all natural solution that satisfy $x^2+y^2 = 3z^2$
negative_passages.docid: 157710
negative_passages.text: $x^{3}-x-2=0$ how do I find the roots?
negative_passages.docid: 48127
negative_passages.text: Are there infinitely many $A\in \mathbb{C}^{2 \times 2}$ satisfying $A^3 = A$?
negative_passages.docid: 71361
negative_passages.text: Prove that $x^x+y^y=z^z$ doesn't have integer solutions
negative_passages.docid: 79925
negative_passages.text: Infinitely many values of $x$ that satisfy the equations
negative_passages.docid: 34435
negative_passages.text: Integral solutions to $y^{2}=x^{3}-1$
negative_passages.docid: 137015
negative_passages.text: Solutions for $a^2+b^2+c^2=d^2$
negative_passages.docid: 72023
negative_passages.text: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
negative_passages.docid: 24307
negative_passages.text: Is $x^x=y$ solvable for $x$?
negative_passages.docid: 55119
negative_passages.text: Proof that the coefficients of a polynomial are real
negative_passages.docid: 145558
negative_passages.text: Prove that $2^n+3^n $ is never a perfect square
negative_passages.docid: 104498
negative_passages.text: Show that $x^3y^3(x^3+y^3) \leq 2$
negative_passages.docid: 24837
negative_passages.text: $x^2+1=0$ uncountable many solutions
negative_passages.docid: 144863
negative_passages.text: Finding integer solutions to $y^2=x^3-2$