In number theory, Bézout's identity, named after Étienne Bézout, is a linear diophantine equation. It states that if a and b are integers with greatest common divisor d, then there exist integers x and y such that Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Étienne Bézout (March 31, 1730 - September 27, 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Basses-Loges (near Fontainbleau), France. ... A linear equation in algebra is an equation which is constructed by equating two linear functions. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (GCF) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. ...

ax + by = d.

Numbers x and y as above can be determined with the extended Euclidean algorithm, but they are not uniquely determined. The extended Euclidean algorithm is a version of the Euclidean algorithm; its input are two integers a and b and the algorithm computes their greatest common divisor (gcd)/Highest Common Factor(HCF), as well as integers x and y such that ax + by = gcd(a, b). ...

For example, the greatest common divisor of 12 and 42 is 6, and we can write

(-3)·12 + 1·42 = 6

and also

4·12 + (-1)·42 = 6.

The greatest common divisor d of a and b is in fact the smallest positive integer that can be written in the form ax + by.

Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID). That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b, then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra+Rb is principal and indeed is equal to Rd. In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, the term ideal has multiple meanings. ...

To confirm: In some credible books, this identity has been attributed to French mathematician Claude Gaspard Bachet de Méziriac. Claude Gaspard Bachet de Méziriac (October 9, 1581 - February 26, 1638) was a French mathematician born in Bourg-en-Bresse. ...

External link

• Online calculator (http://wims.unice.fr/wims/wims.cgi?module=tool/arithmetic/bezout.en) of Bézout's identity.