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

In summary, the conversation is about finding a solution for the equation d^2-ab=e^2-bc=f^2-ac in the natural numbers, where not all variables are equal. The suggestion is to let a, b, and c be any square numbers and d, e, and f be defined so that the differences are zero. It is also mentioned that if non-zero differences are desired, the solution will be more complicated. Additionally, it is suggested that any solution for the equation would result in a tetrahedron with side lengths a,b,c,d,e,f and a sphere located within touching all sides.
  • #1
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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  • #2
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.
 
  • #3
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.
 

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

What is an equation in natural numbers?

An equation in natural numbers is a mathematical statement that uses only positive integers (whole numbers) as its variables and coefficients. It is commonly used to represent relationships between quantities or to solve problems involving counting or measuring.

How is an equation in natural numbers different from other types of equations?

An equation in natural numbers is unique because it only uses positive integers, whereas other types of equations can involve negative numbers, fractions, or decimal numbers. It also typically involves counting or measuring, rather than representing abstract concepts or variables.

Can equations in natural numbers be solved?

Yes, equations in natural numbers can be solved by finding the value of the variable that makes the equation true. This can be done using basic arithmetic operations such as addition, subtraction, multiplication, and division.

What is the purpose of using equations in natural numbers?

The purpose of using equations in natural numbers is to represent and solve real-world problems involving counting or measuring. It allows scientists and mathematicians to model and analyze relationships between quantities in a concrete and precise way.

How are equations in natural numbers used in scientific research?

Equations in natural numbers are used extensively in scientific research, particularly in fields such as physics, chemistry, biology, and economics. They are used to describe and predict natural phenomena and to develop mathematical models for complex systems. Scientists also use equations in natural numbers to analyze data and test hypotheses in their experiments.

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