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I.Statement of Bézout’s theorem:
Let a and b be two non-zero natural numbers.
To say that a and b are prime is equivalent to saying that there are two integers u and v such that .
Demonstration:
1.suppose there are two integers u and v such that au + bv = 1 and prove that a and b are prime to each other.
We note
divides a and b so
divides au + bv.
Since au + bv = 1, = 1 and a and b are prime to each other.
2.suppose that a and b are prime and show that 1 is written in the form au + bv.
Let be the set of numbers in the form au + bv with
and
.
The set is not empty because for u = 1 and v = 0,
.
Thus contains strictly positive integers, and, among them, one smaller than all the others.
Note this smallest element.
The euclidean division of a by m is written with
or .
Thus .Now m is the smallest strictly positive integer of
so r = 0.
Thus m divides a.We show in the same way that m divides b.
Since a and b are prime, m=1 and .
In practice, how to find u and v?
To determine the coefficients, we use Euclid’s algorithm.
Let’s give an example.
We look for a pair (x;y) of relative integers such that 89x+41y=1 (1).
89 and 41 are prime to each other so there are two relative integers x and y satisfying (1).
Let a=89 and b=41.
so
.
so
.
so
.
Let .
Thus is a solution of (1).
II. a new characterization of the PGCD
a and b are two non-zero natural numbers.To say that is the
is equivalent to saying that
is a divisor of a and b and there are two relative integers u and v such that
.
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