The properties of prime numbers

The set is the set of prime numbers:

We call a prime number, a number which has only itself and as a divisor.

Likewise, we will say that two numbers are coprime if their unique common divisor is .

Two prime numbers are coprime

Two prime numbers are coprime.

Breakdown in prime factors

All natural number uniquely decomposes into a prime factors product.

Every number higher than 2 owns at least one prime divisor

All natural number has at least one prime divisor.

Every non-prime number higher than 4 owns at least one strict divisor

All non-prime natural number has at least one strict divisor such as .

Coprimity link between a prime number and an integer

Euclid's lemma

Euclid's lemma corollary

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Demonstrations

Two prime numbers are coprime

Let be two prime numbers.

So:

Their only common divisor is .

Then,

The reciprocal is not true.

For example:

And yet these numbers are not prime.

Breakdown in prime factors

Let be a natural number with .

This number admits a finite number of divisors.

1. if is prime

There is only one factor.

2. if is not prime

We know that has at least one prime divisor.

1. - if is not prime

On recommence :

2. - if is not prime

We carry out this process until the last divisor which will necessary be prime.




We could possibly come across the same prime numbers several times in a row.

And finally,

All natural number uniquely decomposes into a prime factors product.

Every number higher than 2 owns at least one prime divisor

Let be a natural number with .

Two cases arise:

1. is prime

Then, .

It has at least one prime divisor.

2. is not prime

has at least one strict divisor.

Let be the smallest stricit divisor of , is necessarily prime, because if it were not, it would have a divisor lower than itself which would divide , and would not be the smallest divisor of .

And finally,

All natural number has at least one prime divisor.

Every non-prime number higher than 4 owns at least one strict divisor

Let be a non-prime integer with , and a strict divisor of .

So,

By multiplying both members by ,

And finally,

All non-prime natural number has at least one strict divisor such as .

Coprimity link between a prime number and an integer

Let be a prime number and an integer.

If then . Hence the fact that:

Reciprocal

If , as divides only itself and , so .

And finally,

Euclid's lemma

Let be a prime number, two integers.

If , then two cases arise:

1. divides

Alors, le théorème est vrai

2. does not divide

So, the coprimity link between a prime number and an integer tells us that as , so .

Now, with Gauss' theorem:

Which allows us to say that divides .

And finally,

Euclid's lemma corollary

Let be three prime numbers.

If , we saw above that with Euclid's lemma, we do have this:

Let see what happens in both cases.

1. divides

is prime so its only divisors are and .

But, by hypothesis so it is a prime number, so:

2. divides

These are the same hypotheses for , therefore:

And finally,