solve x^n +y^n = z^n in csharp

To solve the equation x^n + y^n = z^n in C#, we can use a brute force approach to check all possible values of x, y, and z within a given range. However, it's important to note that this equation is a special case of Fermat's Last Theorem, which states that there are no whole number solutions to the equation a^n + b^n = c^n for n > 2.

Therefore, we cannot find integer solutions for this equation using our typical numerical methods. Instead, we can implement an algorithm that searches for a counter-example to the theorem, which was famously proven by Andrew Wiles in 1994.

In C#, we can write an implementation of this algorithm as follows:

main.cs
using System;

public class FermatSolver {
    public static bool CheckFermat(int x, int y, int z, int n) {
        int lhs = (int) Math.Pow(x, n) + (int) Math.Pow(y, n);
        int rhs = (int) Math.Pow(z, n);

        return lhs == rhs;
    }

    public static void Main() {
        int n = 3; // Set value of n for the equation
        bool fermatIsTrue = true;

        // Search for counter-examples
        for (int x = 1; fermatIsTrue && x < 100; x++) {
            for (int y = 1; fermatIsTrue && y < 100; y++) {
                for (int z = 1; fermatIsTrue && z < 100; z++) {
                    if (CheckFermat(x, y, z, n)) {
                        fermatIsTrue = false;
                        Console.WriteLine($"Counter-example found: {x}^{n} + {y}^{n} = {z}^{n}");
                    }
                }
            }
        }

        if (fermatIsTrue) {
            Console.WriteLine("No counter-example found - Fermat's Last Theorem is true!");
        }
    }
}
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This program sets n = 3, and then checks all possible values of x, y, and z up to 100 for a counter-example to the theorem. If a solution is found, the program will print the values of x, y, and z that satisfy the equation.

However, as previously noted, we know that there are no integer solutions to this equation for n > 2. Therefore, this program will not be able to find a solution and will return "No counter-example found - Fermat's Last Theorem is true!".

Note: In order to find numerical solutions to the equation x^n + y^n = z^n for n > 2, we must use more advanced mathematical techniques, such as elliptic curves or modular forms, which are beyond the scope of this answer.