From 0e9d69b8d9ece72a1af266d5bb2473f9fee604d8 Mon Sep 17 00:00:00 2001
From: jxu <7989982+jxu@users.noreply.github.com>
Date: Sun, 13 Oct 2024 21:07:36 -0400
Subject: [PATCH 1/3] =?UTF-8?q?linear-diophantine:=20add=20B=C3=A9zout's?=
=?UTF-8?q?=20lemma=20and=20slightly=20reformat=20mathjax?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
---
src/algebra/linear-diophantine-equation.md | 16 ++++++++++++----
1 file changed, 12 insertions(+), 4 deletions(-)
diff --git a/src/algebra/linear-diophantine-equation.md b/src/algebra/linear-diophantine-equation.md
index 047067c59..2364e99fa 100644
--- a/src/algebra/linear-diophantine-equation.md
+++ b/src/algebra/linear-diophantine-equation.md
@@ -23,14 +23,22 @@ In this article, we consider several classical problems on these equations:
A degenerate case that need to be taken care of is when $a = b = 0$. It is easy to see that we either have no solutions or infinitely many solutions, depending on whether $c = 0$ or not. In the rest of this article, we will ignore this case.
+## Bézout's lemma
+
+Bézout's lemma (also called Bézout's identity) is a useful result that can be used to understand and prove the following analytic solution.
+
+> Let $g = \gcd(a,b)$. Then there exist integers $x,y$ such that $ax + by = g$.
+>
+> Moreover, $g$ is the least such positive integer that can be written as $ax + by$; all integers of the form $ax + by$ are multiples of $g$.
+
## Analytic solution
When $a \neq 0$ and $b \neq 0$, the equation $ax+by=c$ can be equivalently treated as either of the following:
-\begin{gather}
-ax \equiv c \pmod b,\newline
-by \equiv c \pmod a.
-\end{gather}
+\begin{align}
+ax &\equiv c \pmod b \\
+by &\equiv c \pmod a
+\end{align}
Without loss of generality, assume that $b \neq 0$ and consider the first equation. When $a$ and $b$ are co-prime, the solution to it is given as
From 5617b117b8888e7e28ba27d53f63446c7285c9e0 Mon Sep 17 00:00:00 2001
From: jxu <7989982+jxu@users.noreply.github.com>
Date: Sun, 13 Oct 2024 21:29:12 -0400
Subject: [PATCH 2/3] linear-diophantine: move identity
---
src/algebra/linear-diophantine-equation.md | 14 ++++++--------
1 file changed, 6 insertions(+), 8 deletions(-)
diff --git a/src/algebra/linear-diophantine-equation.md b/src/algebra/linear-diophantine-equation.md
index 2364e99fa..852d0f255 100644
--- a/src/algebra/linear-diophantine-equation.md
+++ b/src/algebra/linear-diophantine-equation.md
@@ -23,14 +23,6 @@ In this article, we consider several classical problems on these equations:
A degenerate case that need to be taken care of is when $a = b = 0$. It is easy to see that we either have no solutions or infinitely many solutions, depending on whether $c = 0$ or not. In the rest of this article, we will ignore this case.
-## Bézout's lemma
-
-Bézout's lemma (also called Bézout's identity) is a useful result that can be used to understand and prove the following analytic solution.
-
-> Let $g = \gcd(a,b)$. Then there exist integers $x,y$ such that $ax + by = g$.
->
-> Moreover, $g$ is the least such positive integer that can be written as $ax + by$; all integers of the form $ax + by$ are multiples of $g$.
-
## Analytic solution
When $a \neq 0$ and $b \neq 0$, the equation $ax+by=c$ can be equivalently treated as either of the following:
@@ -59,6 +51,12 @@ y = \frac{c-ax}{b}.
## Algorithmic solution
+**Bézout's lemma** (also called Bézout's identity) is a useful result that can be used to understand and prove the following solution.
+
+> Let $g = \gcd(a,b)$. Then there exist integers $x,y$ such that $ax + by = g$.
+>
+> Moreover, $g$ is the least such positive integer that can be written as $ax + by$; all integers of the form $ax + by$ are multiples of $g$.
+
To find one solution of the Diophantine equation with 2 unknowns, you can use the [Extended Euclidean algorithm](extended-euclid-algorithm.md). First, assume that $a$ and $b$ are non-negative. When we apply Extended Euclidean algorithm for $a$ and $b$, we can find their greatest common divisor $g$ and 2 numbers $x_g$ and $y_g$ such that:
$$a x_g + b y_g = g$$
From b1ba5990e50b1c63121750a8891218c8c4636b8f Mon Sep 17 00:00:00 2001
From: jxu <7989982+jxu@users.noreply.github.com>
Date: Mon, 14 Oct 2024 13:45:38 -0400
Subject: [PATCH 3/3] =?UTF-8?q?linear-diophantine:=20reword=20B=C3=A9zout?=
=?UTF-8?q?=20and=20explain=20all=20solutions?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
---
src/algebra/linear-diophantine-equation.md | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/src/algebra/linear-diophantine-equation.md b/src/algebra/linear-diophantine-equation.md
index 852d0f255..3f4ede86c 100644
--- a/src/algebra/linear-diophantine-equation.md
+++ b/src/algebra/linear-diophantine-equation.md
@@ -51,7 +51,7 @@ y = \frac{c-ax}{b}.
## Algorithmic solution
-**Bézout's lemma** (also called Bézout's identity) is a useful result that can be used to understand and prove the following solution.
+**Bézout's lemma** (also called Bézout's identity) is a useful result that can be used to understand the following solution.
> Let $g = \gcd(a,b)$. Then there exist integers $x,y$ such that $ax + by = g$.
>
@@ -125,7 +125,7 @@ $$y = y_0 - k \cdot \frac{a}{g}$$
are solutions of the given Diophantine equation.
-Moreover, this is the set of all possible solutions of the given Diophantine equation.
+Since the equation is linear, all solutions lie on the same line, and by the definition of $g$ this is the set of all possible solutions of the given Diophantine equation.
## Finding the number of solutions and the solutions in a given interval
pFad - Phonifier reborn
Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.
Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.
Alternative Proxies:
Alternative Proxy
pFad Proxy
pFad v3 Proxy
pFad v4 Proxy