It seems like you're asking to solve the equation:

$x^n + y^n = z^n$

This equation is a form of Fermat's Last Theorem, which states that there are no whole number solutions (positive integers) for $x$, $y$, and $z$ when $n > 2$. The theorem was proven by Andrew Wiles in 1994.

For $n = 2$ (the Pythagorean theorem case):

The equation becomes:

$x^2 + y^2 = z^2$

This has many integer solutions, such as:

$3^2 + 4^2 = 5^2$ or $5^2 + 12^2 = 13^2$

But for $n > 2$, Fermat's Last Theorem tells us that there are no integer solutions to the equation.

Would you like to discuss the theorem further or focus on a specific case (like when $n = 2$)?

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Here are 5 related questions:

1. What is Fermat's Last Theorem and its significance in mathematics?
2. Can you provide examples of Pythagorean triples?
3. How did Andrew Wiles prove Fermat's Last Theorem?
4. What are the general solutions to Diophantine equations?
5. How do exponents work in modular arithmetic?

Tip: When dealing with equations involving exponents, it's important to note the behavior of solutions for small values of $n$ before exploring the more complex cases.