query: Find all $x,y,z \in \mathbb{N}$ that $x^xy^y=z^z$
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: 119297
negative_passages.text: Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$
negative_passages.docid: 62980
negative_passages.text: How do you factorise $x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3$?
negative_passages.docid: 114146
negative_passages.text: Polynomial in $n$ variables
negative_passages.docid: 1071
negative_passages.text: Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)
negative_passages.docid: 35066
negative_passages.text: Nature and number of solutions to $xy=x+y$
negative_passages.docid: 21891
negative_passages.text: If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
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: 97411
negative_passages.text: If $x^n=y^n$ and $n$ is odd then $x=y$
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: 21679
negative_passages.text: Solving $x^3-y^3=xy+61$ in integers
negative_passages.docid: 145287
negative_passages.text: Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$.
negative_passages.docid: 123159
negative_passages.text: Solutions of the diophantine equation $x+y+z=xyz$?
negative_passages.docid: 9110
negative_passages.text: Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
negative_passages.docid: 156356
negative_passages.text: Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
negative_passages.docid: 117001
negative_passages.text: Find all solutions for $z^3 = \overline{z}$
negative_passages.docid: 65218
negative_passages.text: Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$