Search for a tool
Bezout's Identity

Tool to compute Bezout coefficients. The Bezout Identity proves that it exists solutions to the equation a.u + b.v = PGCD(a,b).

Results

Bezout's Identity -

Tag(s) : Arithmetics

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Bezout's Identity tool, so feel free to write! Thank you !

# Bezout's Identity

## Bezout Identity Calculator

Tool to compute Bezout coefficients. The Bezout Identity proves that it exists solutions to the equation a.u + b.v = PGCD(a,b).

### What is Bezout Identity?

The Bachet-Bezout identity is defined as : if $a$ and $b$ are two integers and $d$ is their GCD (greatest common divisor), then it exists $u$ and $v$, two integers such as $au + bv = d$.

Example: $a=12$ and $b=30$, gcd $(12, 30) = 6$. There are multiple solutions to $u$ and $v$ such as $12u + 30v = 6$, such as : $$12 \times -2 + 30 \times 1 = 6$$

The dCode Bezout coefficients calculator gives only one solution.

### What are Bezout coefficients?

The Bézouts coefficients are the values $u$ and $v$.

### How to calculate values for Bézout Identity?

The dCode program uses the extended GCD algorithm. $a$ and $b$ are two non-zero positive integers.

The algorithm of dCode consists of a sequence of Euclidean divisions for finding the Bezout coefficients (and also the GCD).

### How to code Bézout Identity in pseudo-code?

A source code for the identity of Bezout would be similar to this pseudo-code:

Initialization r = a, r' = b, u = 1, v = 0, u' = 0 and v' = 1 While (r' != 0) q = (int) r/r' rs = r, us = u, vs = v, r = r', u = u', v = v', r' = rs - q*r', u' = us - q*u', v' = vs - q*v' End While Return (r, u, v)

## Source code

dCode retains ownership of the online 'Bezout's Identity' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Bezout's Identity download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!