query: Diophantine equation: $2 a^2 + 2 b^2 = c^2 + d^2$
positive_passages.docid: 150085
positive_passages.text: Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$
negative_passages.docid: 132098
negative_passages.text: Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
negative_passages.docid: 71756
negative_passages.text: diophantine equation: $x^2 +y^2 = z^n$
negative_passages.docid: 119109
negative_passages.text: Integer solutions to $ n^2 + 1 = 2 \times 5^m$
negative_passages.docid: 60842
negative_passages.text: if $a^2+b^2=c^2$ then $gcd(a,b)=gcd(a,c)=gcd(b,c)$.
negative_passages.docid: 47001
negative_passages.text: Cubic diophantine equation
negative_passages.docid: 72023
negative_passages.text: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
negative_passages.docid: 106271
negative_passages.text: Solutions of Diophantine equations in Natural numbers
negative_passages.docid: 97057
negative_passages.text: Show $a^2+b^2+c^2=1$ $\implies$ $ \dfrac {b^2c^2}{1+a^2} +\dfrac {c^2a^2}{1+b^2}+\dfrac {a^2b^2}{1+c^2} \le \dfrac 14 $
negative_passages.docid: 134159
negative_passages.text: Smallest value of $a^2 + b^2 + c^2+ d^2$, given values for $(a+b)(c+d)$, $(a+c)(b+d)$, and $(a+d)(b+c)$
negative_passages.docid: 136856
negative_passages.text: Diophantine equation with a prime number
negative_passages.docid: 33105
negative_passages.text: Find the value of $a^2-b^2+c^2$
negative_passages.docid: 147562
negative_passages.text: Sum of squares diophantine equation
negative_passages.docid: 63194
negative_passages.text: Find all integer solutions to $x^2-2y^2=1$
negative_passages.docid: 50772
negative_passages.text: Solving a Linear Diophantine Equation
negative_passages.docid: 46028
negative_passages.text: Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.
negative_passages.docid: 99739
negative_passages.text: The diophantine equation $x^2+y^2=3z^2$