Diophantine equation and squares

[secondprime]

The Diophantine equation below,

$$ x_0^{2} - (x_1^{2}+x_2^{2}+x_3^{2}+x_4^{2}+x_5^{2}+x_6^{2}+x_7^{2}+x_8^{2})=1$$

1. Does above equation have any specific name?
2. What are the solutions(a formula)??
3. in the case,$$x_8^{2}=0$$ , does anything special happen??
4. What is the general way to approach/solve these kind of equation??

Any kind of comment would help!

 

[jedishrfu]

I think its similar to the Pythagorean formula in a sense:

Here is a discussion on Diophantine equations and type:

http://en.wikipedia.org/wiki/Diophantine_equation

and here a Wolfram MathWorld:

http://mathworld.wolfram.com/DiophantineEquation.html