query: Prove that $ x^xy^y=z^z $ has infinite integral solutions
positive_passages.docid: 23866
positive_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: 141852
negative_passages.text: Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$
negative_passages.docid: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
negative_passages.docid: 86211
negative_passages.text: If $x+y+z \geq xyz$ Prove that $x^2 + y^2 + z^2 \geq xyz$.
negative_passages.docid: 51417
negative_passages.text: Solve: $\overline{z}=z^n$
negative_passages.docid: 52103
negative_passages.text: Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
negative_passages.docid: 112390
negative_passages.text: Solutions of $x^2 \equiv 1 \pmod N$
negative_passages.docid: 28148
negative_passages.text: Are there any simple ways to see that $e^z-z=0$ has infinitely many solutions?
negative_passages.docid: 103020
negative_passages.text: Find all integral solutions of $ x^4 + y^4 + z^4 -w^4 = 1995 $
negative_passages.docid: 90261
negative_passages.text: Find all solutions of the equation $n^m=x^2+py^2$ which satisfy the following properties
negative_passages.docid: 75501
negative_passages.text: Proof using Cauchy Integral Theorem
negative_passages.docid: 41383
negative_passages.text: Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
negative_passages.docid: 20626
negative_passages.text: Prove that $x^y < y^x$
negative_passages.docid: 91196
negative_passages.text: Complex Integration of $\int_0^\infty e^{-ax}\cos(bx)\,dx$
negative_passages.docid: 129584
negative_passages.text: How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$