How to Solve Quadratic Diophantine Equations in Natural Numbers?

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Homework Help Overview

The discussion revolves around solving quadratic Diophantine equations, specifically the equations x² + y² = z² - 1 and x² + 3y² = z², with the requirement that x, y, and z are natural numbers.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the feasibility of solving three variables with only two equations and question the implications of the natural numbers constraint. There is also a discussion about the interpretation of "solving" in this context, whether it refers to finding specific solutions or understanding the nature of the equations.

Discussion Status

Some participants have provided insights into the nature of the equations and the challenges involved in finding solutions. There is acknowledgment of the complexity of the problem, and while links to resources have been shared, no consensus on a specific approach has emerged.

Contextual Notes

Participants note the potential confusion arising from the requirement for natural numbers and the implications this has on the interpretation of the equations.

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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