AN ANALYTIC APPROACH OF SOME CONJECTURES RELATED TO DIOPHANTINE EQUATIONS

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


The first sequences
We have

LEMMA 1
Let now the following equation

So
The proof of the theorem or the application the sequences and series We will consider firstly that x>y Also And

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We will study now the convergence of the series. As is convergent and is convergent. It means that The reasoning is the same for x<y.
We have

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We have then

Second sequences
(6) x, y, z, a and n are positive integers. Thus (10)

First conclusion
The new equations allow to build sequences and series that leads to test the impossibility of the resolution of an equation. If they are a consequence of some Diophantine equations, they remain an intellectual building.

The generalized sequences
Now, we will generalize the results. Let the following equation

LEMMA 6
We will define the sequences Which implies

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The reasoning is available until infinity. Then

Proof of lemma 7
By traditional induction, it is verified for j=1, we suppose that (P) is true for j, so And it is true for j+1.

LEMMA 8
The equation (E) leadss to an impossibility, effectively, if we pose Which leads, we will see it, to x=y Because they are coprime. Now, the question is : why did we propose solutions for 38 BMSA Volume 1

Let us pose
It is the expression for the exponent (i-1). If there are solutions for the exponent (i-1), there will be solutions for the exponent i(i-1). It is not true for i, because of the exponent -(i-1) in the expression (P).

Second conclusion
The sequences and the series as we defined them have several applications in several diophantine equations, for example Fermat, Beal, Erdos, we saw the generalized equation (1), but there are many others like Pillai, Smarandache, Catalan…

Other sequences
Now, let the equation (4) We will build sequences

And
The process is available until infinity, for j And The expressions are We prove it by induction, like we did for rational sequences So So the only solution is x=y=0

Conclusion
It appeared since the beginning, before the change of the data, that the equations contain a symmetry between x and y. effectively, we found u=x+y. We broke the symmetry by changing the equation in two equations .The conclusion is that the equation (1) leads always to an impossibility which is x=y. It is the case of Fermat-Catalan or Erdos equations. The cause is the undecidability of some conjectures related to Diophantine equations.