Can a Tetrahedron be Constructed with the Given Equation in Natural Numbers?

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SUMMARY

The discussion centers on the equation d² - ab = e² - bc = f² - ac, seeking solutions in natural numbers where not all variables are equal. The variables a, b, and c are specified as square numbers, while d, e, and f are derived to ensure the differences equal zero. The conversation highlights the geometric interpretation of the solution, indicating that valid solutions correspond to the side lengths of a tetrahedron capable of enclosing a sphere that touches all sides. The complexity increases when non-zero differences are introduced.

PREREQUISITES
  • Understanding of natural numbers and their properties
  • Familiarity with square numbers and their applications
  • Basic knowledge of geometric concepts, particularly tetrahedrons
  • Elementary algebra for manipulating equations
NEXT STEPS
  • Research methods for solving Diophantine equations
  • Explore geometric properties of tetrahedrons and inscribed spheres
  • Learn about numerical methods for finding integer solutions
  • Investigate the implications of non-zero differences in similar equations
USEFUL FOR

Mathematicians, geometry enthusiasts, and students interested in number theory and geometric constructions will benefit from this discussion.

disregardthat
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Hi, I was wondering if anyone could find a solution to this:

[tex]d^2-ab=e^2-bc=f^2-ac[/tex]

in the natural numbers where not all variables are equal. I don't know how to make a computer program, but if it takes little time, I would really appreciate if I could have a solution to it.
 
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Let a, b, and c be any square numbers, and let d, e, and f be defined so that the differences are zero. The rest is trivially obvious.

If you want non-zero differences it will be more complicated.
 
Of course =) thanks (cant believe i didn't think of that)

Any solution a,b,c,d,e,f gives a tetrahedron of side lengths a,b,c,d,e,f where a sphere may be located within touching all sides - if my calculations are correct.
 

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