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

ax + by = d.

Numbers x and y as above can be determined with the extended Euclidean algorithm, but they are not uniquely determined: $a(x\; -\; kb)\; +\; b(y\; +\; ka)\; =\; ax\; -\; kba\; +\; by\; +\; kba\; =\; ax\; +\; by$ for every a, b, x, y, and k. So, letting k range over the integers and setting $x\; ^\backslash prime\; =\; x\; -\; kb$, $y^\backslash prime\; =\; y\; +\; ka$, we have $ax^\backslash prime\; +\; by^\backslash prime\; =\; ax\; +\; by$.

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

12x + 42y = 6.

with some of the solutions being

(-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. An integral domain in which Bézout's identity holds is called a Bézout domain.

To confirm: In some credible books, this identity has been attributed to French mathematician Claude Gaspard Bachet de Méziriac.

External links

---

• Online calculator of Bézout's identity.
• Bezout's Identity for Gaussian Integers, Fermat's Last Theorem Blog covers topics in the history of Fermat's Last Theorem.

Diophantine equations

Lemma von Bézout | Identitat de Bézout | Identidad de Bézout | Identité de Bézout | Identità di Bézout | Теорема Безу | 貝祖等式