How to Solve Quadratic Diophantine Equations in Natural Numbers?

[oszust001]

How can i solve that equation:
x^2 + y^2 = z^2-1 or x^2 + 3y^2 = z^2?

 

[madah12]

I don't think you can solve 3 variables with two equations but I am not sure.

 

[Tide]

I don't think you can solve 3 variables with two equations but I am not sure.

It depends on what you mean by "solve." The OP might be looking for a curve in space or a surface or some such. However, the "natural numbers" part confuses the issue.

 

[Mentallic]

I'm sure the OP means to find the set of solutions for those equations where x,y,z are all natural numbers.

 

[HallsofIvy]

Those are quadratic Diophantine equations. I don't know much about them myself by here is a link:http://mathworld.wolfram.com/DiophantineEquation2ndPowers.html

 

[D H]

Those are quadratic Diophantine equations. I don't know much about them myself by here is a link:http://mathworld.wolfram.com/DiophantineEquation2ndPowers.html

Equations 6 to 10 of that article provide an explicit algorithm for solving exactly the kinds of problems specified in the original post.