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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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