51 lines
1.3 KiB
Markdown
51 lines
1.3 KiB
Markdown
#Math #NT
|
||
|
||
# Statement
|
||
|
||
Let $x \in \mathbb{Z}$, $y \in \mathbb{Z}$, $x \neq 0$, $y \neq 0$, and $g = \gcd(x, y)$. Bezout's Identity states that $\alpha \in \mathbb{Z}$ and $\beta \in \mathbb{Z}$ exists when:
|
||
|
||
$$
|
||
\alpha x + \beta y = g
|
||
$$
|
||
|
||
Furthermore, $g$ is the least positive integer able to be expressed in this form.
|
||
|
||
# Proof
|
||
|
||
## First Statement
|
||
|
||
Let $x = gx_1$ and $y = gy_1$, and notice $\gcd(x_1, y_1) = 1$ and $\operatorname{lcm} (x_1, y_1) = x_1 y_1$.
|
||
|
||
Since this is true, the smallest integer $\alpha$ for $\alpha x_1 \equiv 0 \mod y$ is $a = y_1$.
|
||
|
||
For all integers $0 \leq a, b < y_1$, $ax_1 \not\equiv bx_1 \mod y$. (If not, we get $|b - a| > y_1$, which is contradictory). Thus, by pigeonhole principle, there exists $\alpha$ such that $\alpha x_1 \equiv 1 \mod y_1$.
|
||
|
||
Therefore, there is an $\alpha$ such that $ax_1 - 1 \equiv 0 \mod y_1$, and by extension, there exists an integer $\beta$ such that:
|
||
|
||
$$
|
||
\alpha x_1 - 1 = -\beta y_1 \\
|
||
\alpha x_1 + \beta y_1 = 1
|
||
$$
|
||
|
||
By multiplying by $g$:
|
||
|
||
$$
|
||
\alpha x + \beta y = g
|
||
$$
|
||
|
||
## Second Statement
|
||
|
||
To prove $g$ is minimum, let’s consider another positive integer $g\prime$:
|
||
|
||
$$
|
||
\alpha\prime x + \beta\prime y = g\prime
|
||
$$
|
||
|
||
Since all values are a multiple of $g$:
|
||
|
||
$$
|
||
0 \equiv \alpha \prime x + \beta \prime x \mod g \\
|
||
0 \equiv g\prime \mod g
|
||
$$
|
||
|
||
Since $g$ and $g\prime$ are positive integers, $g\prime \geq g$. |