query: proof - if $x^2 + y^2 + z^2 = 2xyz$ then $x = y = z = 0$
positive_passages.docid: 72023
positive_passages.text: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
negative_passages.docid: 31317
negative_passages.text: If we know $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, how to find $x^4+y^4+z^4$?
negative_passages.docid: 89637
negative_passages.text: Proof that in a field $x=0$ is equivalent to $x=-x$
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 152167
negative_passages.text: Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.
negative_passages.docid: 71361
negative_passages.text: Prove that $x^x+y^y=z^z$ doesn't have integer solutions
negative_passages.docid: 59239
negative_passages.text: For non-zero real $x$, $y$, $z$ and complex $a$, $b$, $c$ with $|a|=|b|=|c|$, if $x+y+z=0=ax+by+cz$, then $a=b=c$
negative_passages.docid: 90549
negative_passages.text: If $z^2 = z_1^2+z_2^2+z_3^2$ then $|\Re(z)| \le|\Re(z_1)|+|\Re(z_2)|+|\Re(z_3)|$
negative_passages.docid: 155757
negative_passages.text: Number theory: $x^y + 1 = y^x$
negative_passages.docid: 33082
negative_passages.text: Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
negative_passages.docid: 29151
negative_passages.text: Equation with Vieta Jumping: $(x+y+z)^2=nxyz$.
negative_passages.docid: 5595
negative_passages.text: $x^y = y^x$ for integers $x$ and $y$
negative_passages.docid: 41689
negative_passages.text: Find all solutions: $x^2 + 2y^2 = z^2$
negative_passages.docid: 75387
negative_passages.text: A proof that $1=2$. May I know why it’s false?
negative_passages.docid: 87665
negative_passages.text: Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$
negative_passages.docid: 86285
negative_passages.text: Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
negative_passages.docid: 71756
negative_passages.text: diophantine equation: $x^2 +y^2 = z^n$