TITLE: Is there an algorithm to solve quadratic Diophantine equations?
QUESTION [19 upvotes]: I was asked two questions related to Diophantine equations.

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.
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

REPLY [3 votes]: S. Carnahan (Sep 24 '13 at 18:33)  showed parametrization by cosets of congruence subgroups
of SL(2,Z). What is missing is parametrization of SL(2,Z) and its congruence subgroups  which can be found in 
L.N. Vaserstei. Polynomial parametrization  for the solutions of Diophantine equations and
arithmetic groups , Annals of Math.  171:2 (March of 2010),   979–1009. 
MR2630059. Zbl 05712747