query: How many integral triplets ($x, y, z$) satisfy the equation $x^2 + y^2 + z^2 = 1855$?
positive_passages.docid: 22973
positive_passages.text: Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.
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: 50760
negative_passages.text: Solutions to $y^2 = x^3 + k$?
negative_passages.docid: 153241
negative_passages.text: Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$
negative_passages.docid: 136933
negative_passages.text: Which $n$ can be expressed as $x^2 − y^2$?
negative_passages.docid: 1001
negative_passages.text: Number of integer solutions of $x^2 + y^2 = k$
negative_passages.docid: 30126
negative_passages.text: Linear Diophantine Equations in Three Variables
negative_passages.docid: 70406
negative_passages.text: Find all values x, y and z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes.
negative_passages.docid: 87345
negative_passages.text: How many ways so that $x+y+z\leq n$?
negative_passages.docid: 66161
negative_passages.text: How to solve the equation $n^2 \equiv 0 \pmod{584}$?
negative_passages.docid: 116223
negative_passages.text: Show that the angles satisfy $x+y=z$
negative_passages.docid: 72023
negative_passages.text: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
negative_passages.docid: 143194
negative_passages.text: How would you solve the diophantine $x^4+y^4=2z^2$
negative_passages.docid: 137015
negative_passages.text: Solutions for $a^2+b^2+c^2=d^2$
negative_passages.docid: 41939
negative_passages.text: How many ordered quadruples $(w, x, y, z)$ of non-negative integers are there such that $wxyz = 288$? Why?
negative_passages.docid: 47001
negative_passages.text: Cubic diophantine equation
negative_passages.docid: 152167
negative_passages.text: Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.