TITLE: Second degree Diophantine equations
QUESTION [9 upvotes]: I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know whether one can find all integers satisfying $x^2 + x = y^2 + y + z^2 + z$.

REPLY [3 votes]: There is an algorithm for deciding whether a single degree 2 multivariable polynomial equation has a solution in integers, due to Siegel, Zur Theorie der quadratischen Formen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1972, 21-46.  See also Grunewald and Sigel, On the integer solutions of quadratic equations, J. Reine Angew. Math. 569 (2004), 13-45.  
But the Hasse principle by itself does not give an algorithm: it holds only for rational solutions, not for integer solutions.