X^2+y^2+z^2=nxyz n is natural

Thread starter hadi amiri 4
Start date Dec 7, 2008
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In summary, the Diophantine equation "X^2+y^2+z^2=nxyz n is natural" has been studied for centuries and has applications in fields such as cryptography and coding theory. It is named after the ancient Greek mathematician, Diophantus, and involves finding solutions for a set of variables that satisfy the given equation. The equation has been used in cryptography to create encryption algorithms, particularly in the "knapsack problem". It also has a relationship with Pythagorean triples, where every Pythagorean triple can be a solution to the equation. There are some general rules and patterns for finding solutions, but there is no general algorithm for finding all solutions for any given n. The equation can be extended to

Dec 7, 2008

hadi amiri 4

can anyone give solution to this quation
x^2+y^2+z^2=nxyz n is natural

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Dec 7, 2008

CRGreathouse

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(x, y, z, n) = (1, 1, 1, 3)

Dec 14, 2008

rscosa

Hi,
The answer is described ad a hyper-surface of dimension 2 except if you are talking about a diophantic equation over the integers, please be more especific.

1. What is the significance of the equation "X^2+y^2+z^2=nxyz n is natural" in mathematics?

The equation "X^2+y^2+z^2=nxyz n is natural" is known as the Diophantine equation and is often used in number theory. It is named after the ancient Greek mathematician, Diophantus, and involves finding solutions for a set of variables that satisfy the given equation. It has been studied by mathematicians for centuries and has applications in fields such as cryptography and coding theory.

2. How is the Diophantine equation used in cryptography?

The Diophantine equation has been used in cryptography to create encryption algorithms. In particular, it has been used to develop the "knapsack problem" where the variables in the equation represent the weights of objects in a knapsack. This problem is used in public key cryptography to create secure communication between two parties.

3. What is the relationship between the solutions of the Diophantine equation and Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem (a^2 + b^2 = c^2). Interestingly, every Pythagorean triple can also be a solution to the Diophantine equation "X^2+y^2+z^2=nxyz n is natural". However, not all solutions to the Diophantine equation will be Pythagorean triples.

4. Are there any patterns or rules for finding solutions to the Diophantine equation?

Yes, there are some general rules and patterns for finding solutions to the Diophantine equation. For example, if n is a prime number, then there are infinitely many solutions to the equation. Also, if the equation has solutions for n=1, then it will also have solutions for all other natural numbers. However, there is no general algorithm for finding all solutions to the equation for any given n.

5. Can the Diophantine equation be extended to more than three variables?

Yes, the Diophantine equation can be extended to more than three variables. In general, the equation can be written as a^2+b^2+c^2+...=nxyz... where there are any number of variables on the left side. However, the more variables there are, the more difficult it becomes to find solutions to the equation. This is why the three variable case is the most commonly studied and used in applications.