I was asked two questions related to Diophantine equations.


1. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all solutions but no points which do not satisfy the equation.
2. Is there an algorithm that will determine, given any quadratic $Q(x\_1,\ldots,x\_n)$ as input, all integer points of this quadratic? In the case of existence of solution, there is an algorithm, <https://math.stackexchange.com/questions/181380/second-degree-diophantine-equations/181384#comment418090_181384>



Is there an algorithm to solve quadratic Diophantine equations?
Sierpsinski proved that whenever this diophantine equation does not have a solution for $x$ then $x^2+(x+1)^2$ is prime. It is conjectured also by Sierpinski that there are infinitely many primes of the above form but this still remains open.


So, your question cannot be answered yet. For more details see <http://arxiv.org/pdf/0810.0222.pdf>


