Bezout's Identity


Bézout’s identity, also known as Bézout’s lemma, is a fundamental theorem in elementary number theory. Named after Étienne Bézout, who proved it for polynomials, the identity states the following:

Given two nonzero integers $a$ and $b$, let $d$ be their greatest common divisor (GCD). Then there exist integers $x$ and $y$ such that:

$$ax + by = d$$

In other words, Bézout’s identity guarantees that we can express the GCD of $a$ and $b$ as a linear combination of these integers. Furthermore, the integers $x$ and $y$ are not unique; there are infinitely many pairs that satisfy this equation.
$$ax+by=d$$
$$ax+ab+by-ab=d$$
$$a(x+b)+b(y-a)=d$$
In general $x+nb$ and $y-na$ can be done for any value of $n$. So $x$ and $y$ are not unique.

Key points about Bézout’s identity:
The integers $x$ and $y$ are called Bézout coefficients for the pair $(a, b)$.
These coefficients can be computed using the extended Euclidean algorithm.
Bézout’s identity has applications in various areas of number theory, including modular arithmetic, solving Diophantine equations, and cryptography.

For example, if the GCD of 15 and 6 is 3, we can express 3 as a combination of 15 and 6: 
$3 = 15 \cdot (1) + 6 \cdot (-2)$, with Bézout coefficients 1 and -2.

Remember that Bézout’s identity plays a crucial role in many other theorems in elementary number theory.

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