Bezout's Lemma
Bezout's Lemma states that if and
are nonzero integers and
, then there exist integers
and
such that
. In other words, there exists a linear combination of
and
equal to
.
In particular, if and
are relatively prime then there are integers
and
for which
.
Proof
Let ,
, and notice that
.
Since ,
. So
is smallest positive
for which
. Now if for all integers
, we have that
, then one of those
integers must be 1 from the Pigeonhole Principle. Assume for contradiction that
, and WLOG let
. Then,
, and so as we saw above this means
but this is impossible since
. Thus there exists an
such that
.
Therefore , and so there exists an integer
such that
, and so
. Now multiplying through by
gives,
, or
.
Thus there does exist integers and
such that
.
See also
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