Bézout's identity 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)·12 + 1·42 = 6
and also
- 4·12 + (-1)·42 = 6.
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.
Bézout's identity is named for the 18th century French mathematician Étienne Bézout.
