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# Bézout's identity

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.

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

(-3)ˇ12 + 1ˇ42 = 6

and also

4ˇ12 + (-1)ˇ42 = 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.

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