Bezout's Identity

Bezout’s Identity states that for any two integers ( a ) and ( b ), there exist integers ( x ) and ( y ) such that ( ax + by = \gcd(a, b) ). This theorem is often used to find integer solutions for linear Diophantine equations.

Solution

The Extended Euclidean Algorithm is commonly used to find the integers ( x ) and ( y ) in Bezout’s Identity.

Java

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public class Bezout {
    public static int[] extendedEuclidean(int a, int b) {
        if (b == 0) {
            return new int[] {a, 1, 0};
        }
        int[] result = extendedEuclidean(b, a % b);
        int gcd = result[0];
        int x1 = result[1];
        int y1 = result[2];

        int x = y1;
        int y = x1 - (a / b) * y1;

        return new int[] {gcd, x, y};
    }
}

C++

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#include <tuple>
using namespace std;

tuple<int, int, int> extendedEuclidean(int a, int b) {
    if (b == 0) {
        return {a, 1, 0};
    }
    int gcd, x1, y1;
    tie(gcd, x1, y1) = extendedEuclidean(b, a % b);

    int x = y1;
    int y = x1 - (a / b) * y1;

    return {gcd, x, y};
}

Python

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def extended_euclidean(a, b):
    if b == 0:
        return a, 1, 0
    gcd, x1, y1 = extended_euclidean(b, a % b)

    x = y1
    y = x1 - (a // b) * y1

    return gcd, x, y

In these code snippets, the function extendedEuclidean calculates the integers ( x ) and ( y ) along with ( \gcd(a, b) ). The function uses recursion to solve the problem efficiently. The return values are the gcd, ( x ), and ( y ) in that order.

By running these functions with ( a ) and ( b ) as inputs, you can get ( x ) and ( y ) satisfying Bezout’s Identity.