Diophantine equations

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Sequences, series and numbers generated by diophantine equations and their applications by Jamel Ghanouchi[edit | edit source]

Abstract[edit | edit source]

Our purpose in this article which has been published is to show how much diophantine equations are rich in analytic applications. Effectively, those equations allow to build amazing sequences, series and numbers. The question of the proof of some theorems remains of course, we will see it in this communication. We will make also an allusion to the very known Fermat numbers (). We will see how this problem of the proof is actual and how it can be solved using amazing sequences and series.

The Ghanouchi's sequences[edit | edit source]

Let us begin by Fermat equation (E), it is

with GCD(X,Y)=1

We pose

then

and

If U, X, Y are integers verifying equation (E), then u, x, y, z as defined verify

LEMMA 1[edit | edit source]

Firstly

We pose

and

but

verifying

and

then

or

we pose

and

also

and

which means

and

and

is an integer

is an integer

is an integer

is rational, because

rationals

rational verifying

until infinity. For i

is rational for i>1

is rational for i>1

is rational for i>1

is rational for i>1, and

LEMMA 2[edit | edit source]

and have expressions

Proof of lemma 2[edit | edit source]

By induction

also

it is verified for i=2. We suppose (H) and (H') true for i, then

and

but (H) and (H'), then

and it is true for i+1, also for

and can be written as it follows

but,

the expressions of the sequences become, for

or

LEMMA 3[edit | edit source]

and


LEMMA 4[edit | edit source]

The equations (1) have (2) a constant

THE GHANOUCHI'S THEOREM[edit | edit source]

The only solution of equations

and

is

Proof of Ghanouchi's theorem[edit | edit source]

As

and

if

and

We will give several proofs that is the solution with the series. We recapitulate

x and y are not différent, the initial hypothesis is false there the only solution is

The proofs : we have

is solution of the following equation

Also is solution of

And

Also

Let

But

Hence

We deduce

Also leads to

Hence

We deduce

If we add

The solution is

Another proof : we have

Because

And

Because

And

We deduce

In the infinity

Therefore

Another proof : let

Also

We deduce

And

Thus

And

Also

We have

Let also

And

We deduce

Because

Or

And

But

And

Thus

Hence

Another proof : We have

Let

We have

And

It means

And

But

Thus

And

And

Or

The expression between the parenthesis is not equal to zero, we deduce

Else

We deduce

Thus

The expressions between the parenthesis are not equal, therefore

Else

Hence

Or

The expression between the parenthesis is not equal to zero, we deduce

And, else

And the expression between the parenthesis is not equal to zero, it means that

And

Another proof :

and are particular cases of and which follow

Also

So, we have

And

But

Let

Thus

Or

And

is not equal to zero for , and it is the case, of course, of

therefore the expression between the parenthesis is not equal to zero. We gave the fourth proof that the only solution is

And there are others (see the series).

The Ghanouchi's series[edit | edit source]

As seen

we deduce


then

and

if

if

The applications of the Ghanouchi's series[edit | edit source]

or

and

As we do not know the limit of , then

can be not convergent. But

is convergent.

Also, knowing that tends to zero in the infinity, we can say

is convergent. The limit of

exists and the series are convergent. It means that for x and y integers and for conditions on the exponents like for Fermat equation :

It is confirmed by the fact that the général term of the series tends to zero. Let us prove it. We give two proofs. We must remark that we prove firstly that the following series are convergent, we do not present the proof, here. Let


Also

But

We have

Thus

We have

And

Or

Hance

And

Consequently

Also

We have

Thus

And

And

Or

Hence

Or

Consequently

We deduce

Thus

And

the only solution is , if at least one of the sequences or is constant. And the second proof. Let

let

We Recapitulate

is the only solution of (1) and (2). This result is paradoxal, we remark that we have not put any condition on n, because there are solutions for . The answer is related to Matiasevic theorem which claims that dos not exist an algorithm to prove theorems related to diophantine equations and we gave one : The approach must conduct then to an impossibility. We confirm Matiasevic theorem and prove it because our algorithm is available for n=1. The approach is more important than it appears, it is an answer to problems more general than Fermat theorem or Beal or Fermat-Catalan conjectures. We will try to prove them. The series become

and

and

This development is in fact a test of impossibility. The sequences and series are a consequence of Fermat equation and of other diophantine equations (as we will see). The question now is : why are there solutions for n=2 ? The answer is in the formulas, as seen. It is important to note that for n=1, there are trivial solutions. But, for n=1, lemma 3 allows to write

and

and

It is the expression of , the exponent 2.

The case n=1 conducts to the case n=2 and as there are solutions for n=1, it will be the same for n=2 !

For n=4

the case n=4 is different, in this formula the exponent i-3 does not guarantee the existence of the sequences if

i=2.

Then, the case n=2 is the only exception. The only solution for n>2 is xy(x-y)=0, there is no solution.

Another application is Beal equation. It is

We pose

and

and

it is lemma 1 and, with

the solutions are

or impossible solutions for a>2 et b>2 et c>2.

Another application is the following equation.

We conjecture and prove that there is no solutions for n>i(i-1) and , . We can not know when there are solutions as proved by Matiasevic when one of the exponent is less or equal to i(i-1).

LEMMA 6[edit | edit source]

The solution is


Proof of lemma 6[edit | edit source]

We pose

or

and

it is lemma 1. Its solution is

But, why are not they solutions for n>i(i-1) and  ?

We will generalize the definition of the sequences.

We will define general sequences.

Our goal is to prove that if ,

(), , , ,

are positive integers, then

for the equation

When

there are solutions, for example :

has

and has

and

and has

etc...

We suppose (e) verified and that