query: Prove that $x^2+y^2=z^4$ has infinitely many solutions with $(x,y,z)=1$
positive_passages.docid: 65000
positive_passages.text: prove that $x^2 + y^2 = z^4$ has infinitely many solutions with $(x,y,z)=1$
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: 88760
negative_passages.text: Find the integral solutions to $ x^2+y^2+z^2=x^2y^2$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 63505
negative_passages.text: Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $
negative_passages.docid: 133164
negative_passages.text: Rational solutions for $x^2+y^2=3$
negative_passages.docid: 13572
negative_passages.text: $x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$
negative_passages.docid: 45616
negative_passages.text: Find all solutions of ${\frac {1} {x} } + {\frac {1} {y} } +{\frac {1} {z}}=1$, where $x$, $y$ and $z$ are positive integers
negative_passages.docid: 106896
negative_passages.text: Number of solutions of $x_1+2x_2+\cdots+kx_k=n$?
negative_passages.docid: 50433
negative_passages.text: Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$
negative_passages.docid: 35504
negative_passages.text: The Diophantine equation $x^2 + 2 = y^3$
negative_passages.docid: 140763
negative_passages.text: How to prove that $\sum_{n=1}^{2012} \frac{1}{x_n} = 1 $ has finitely many solutions for positive integer $x_i$?
negative_passages.docid: 158933
negative_passages.text: Rational solutions of $y^2 = x^3 - x$
negative_passages.docid: 147442
negative_passages.text: Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$
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: 55248
negative_passages.text: Positive integral solutions of $3^x+4^y=5^z$
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?