GIVE A GENERAL SOLUTION OF DIOPHENTINE QUADRATIC EQUATION X2+ Y2=Z2

The Diophantine quadratic equation you provided is:

\[
x^2 + y^2 = z^2
\]

This is known as **Pythagoras' equation**, and its solutions are called **Pythagorean triples**. The general integer solutions to this equation can be written in parametric form. These solutions are:

\[
x = m^2 - n^2
\]
\[
y = 2mn
\]
\[
z = m^2 + n^2
\]

where \(m\) and \(n\) are integers, and \(m > n > 0\). These triples \((x, y, z)\) are called **primitive Pythagorean triples** if \(m\) and \(n\) are coprime (i.e., their greatest common divisor is 1) and one of them is even.

### Explanation:

- The expressions for \(x\), \(y\), and \(z\) generate all possible primitive solutions to the equation \(x^2 + y^2 = z^2\).
- Non-primitive solutions (i.e., multiples of the primitive ones) can be generated by multiplying the primitive triples by a common factor \(k\). So the general solution for all Pythagorean triples is given by:

\[
x = k(m^2 - n^2), \quad y = k(2mn), \quad z = k(m^2 + n^2)
\]
where \(k\) is any positive integer.

### Example:

For \(m = 3\) and \(n = 2\), we get the primitive Pythagorean triple:

\[
x = 3^2 - 2^2 = 9 - 4 = 5
\]
\[
y = 2 \cdot 3 \cdot 2 = 12
\]
\[
z = 3^2 + 2^2 = 9 + 4 = 13
\]

Thus, \((5, 12, 13)\) is a Pythagorean triple.

Would you like further details or have any questions about the solution?

### Related Questions:
1. How do you prove that these parametric solutions satisfy \(x^2 + y^2 = z^2\)?
2. Can there be negative values for \(x\), \(y\), or \(z\) in the solutions?
3. What is the significance of coprimeness between \(m\) and \(n\) in generating primitive triples?
4. How do you find the greatest common divisor (GCD) between two numbers?
5. How can we find the number of primitive triples for a given range of \(z\)?

### Tip:
To verify if a set of numbers \((x, y, z)\) forms a Pythagorean triple, just check whether \(x^2 + y^2 = z^2\) holds true.

GIVE A GENERAL SOLUTION OF DIOPHENTINE QUADRATIC EQUATION X2 + Y2 = Z2

Explore the general solution to the Diophantine quadratic equation x^2 + y^2 = z^2, known as Pythagoras' equation. Learn about Pythagorean triples, their parametric formulas, and how to generate primitive and non-primitive triples using integers m and n. Suitable for students in Grades 9-12.

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