Bézout's Lemma is a fundamental result in number theory that describes a special property of the greatest common divisor (GCD) of two integers. It states that the GCD of two integers can always be expressed as a linear combination of those integers.

Let's break down the lemma:

• gcd(a, b): This represents the greatest common divisor of two integers a and b. The GCD is the largest positive integer that divides both a and b without leaving a remainder.
• r * a + s * b = g: This is the core of Bézout's Lemma. It states that there exist integers r and s such that when you multiply r by a, s by b, and add the results, you get the GCD (g) of a and b. The integers r and s are often called Bézout's coefficients.

Why is this important?

Bézout's Lemma is a powerful tool with applications in various areas of mathematics and computer science, including:

• Number Theory: It's used in proofs of other important theorems and in understanding the properties of GCDs.
• Cryptography: It plays a role in certain cryptographic algorithms.
• Computer Science: It's used in algorithms like the Extended Euclidean Algorithm, which efficiently computes the GCD and the coefficients r and s.

Example:

Let's take a = 12 and b = 18. The GCD of 12 and 18 is 6. Bézout's Lemma tells us we can find integers r and s such that 12r + 18s = 6. In this case, we can choose r = -1 and s = 1, because (-1)*12 + (1)*18 = 6.

Key takeaway: Bézout's Lemma guarantees the existence of integers r and s that satisfy the linear combination equation, connecting the GCD of two numbers to a specific linear combination of those numbers.

🧠Demonstration of Bézout's Lemma

https://gist.github.com/viadean/9e476170986d728fb96506a1c5d1a899

Explanation and Output:

1. extended_gcd(a, b) function: This function implements the Extended Euclidean Algorithm. It recursively calculates the GCD and the coefficients r and s. The base case is when a is 0, in which case the GCD is b, and the coefficients are r=0 and s=1. The recursive step uses the modulo operator (%) to reduce the problem to smaller inputs.
2. Demonstration: The code then demonstrates the use of the function with a few examples. It prints the GCD and the coefficients r and s for each pair of numbers. It also verifies that the equation ra + sb = g holds true.

Example Output:

https://gist.github.com/viadean/cefd826f5c597c47bc2a2c517fe3872d

This output shows that for each pair of integers a and b, the code successfully finds the GCD and the Bézout's coefficients r and s that satisfy Bézout's identity. This provides a practical demonstration of Bézout's Lemma.