Wikipedia describes only one way to prove that a homogeneous diophantine equation of degree 2 has no trivial solution (namely by reducing the equation modulo p).
My toolbox for solving homogeneous diophantine equations of degree 2 currently breaks down to the following methods:
- Try out some values for x,y,z. Hope that x,y,z is a non-trivial solution.
- Try out some values for p. Reduce the equation modulo p and hope that you can construct a prove similiar to the corresponding wikipedia page.
These approaches are obviously depending on luck. What other tools do exist?
It is a theorem (the so called Hasse Principle for genus 0 curves) that if $Q(x,y,z) \in \mathbb{Q}[x,y,z]$ is a homogeneous degree $2$ polynomial, then $Q(x,y,z) = 0$ has a solution in $\mathbb{Q}$ if and only if it has a solution in $\mathbb{Q}_p$ for all primes $p$ (roughly speaking (by Hensel's lemma) this is the same as saying there are solutions mod $p$), and a solution in $\mathbb{R}$.
That is to say, this is the only tool one needs when trying to investigate such curves is reduction mod $p$, and the information from $\mathbb{R}$.
Note that in general we do not get nearly as lucky, and a lot of modern number theory is concerned with investigating the Hasse Principle for examples of varieties.