How to Solve Quadratic Diophantine Equations in Natural Numbers?

AI Thread Summary
To solve the quadratic Diophantine equations x^2 + y^2 = z^2 - 1 and x^2 + 3y^2 = z^2 in natural numbers, it's essential to clarify the meaning of "solve." The discussion indicates that the goal is to find sets of natural number solutions for these equations. Quadratic Diophantine equations can be complex, and the original poster may be seeking a geometric interpretation or specific solutions. An external resource is suggested, which provides an explicit algorithm for solving such equations. Understanding the nature of the variables and the constraints of natural numbers is crucial for finding solutions.
oszust001
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How can i solve that equation:
x^2 + y^2 = z^2-1 or x^2 + 3y^2 = z^2?
 
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I don't think you can solve 3 variables with two equations but I am not sure.
 
madah12 said:
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.
 
I'm sure the OP means to find the set of solutions for those equations where x,y,z are all natural numbers.
 
HallsofIvy said:
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.
 

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