Diophantine equations

Sequences, series and numbers generated by diophantine equations and their applications by Jamel Ghanouchi

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Abstract

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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

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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

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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

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  and   have expressions

 

 

Proof of lemma 2

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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

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and

 

 

 


LEMMA 4

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The equations (1) have (2) a constant

 

THE GHANOUCHI'S THEOREM

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The only solution of equations

 

and

 

is

 

Proof of Ghanouchi's theorem

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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

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As seen

 

we deduce

 

 


 

then

 

and

 

if  

 

if  

 

The applications of the Ghanouchi's series

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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

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The solution   is

 


Proof of lemma 6

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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  

 , soit

 

 

and

 

and

 

LEMMA 7

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 ,  , u, v verify

 

and

 

We pose

 

 

 

and

 

 

it implies

 

and

 

 

and

 

The reasoning is available until infinity. Then

 

and

 

and

 

and

 

  are positive  ,  .

LEMMA 8

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(P) is the expression :

 


Proof of lemma 8

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for j=1, it is verified because of the definition of  ,

we suppose (P) true and the expression of   implies, with (P), that

 

or

 

and

 

 

 

then

 

Why are not they solutions for   ?

We suppose

 

the formula becomes

 

 

 

 

It is the expression of   of the exponent i-1.

If we suppose that exist solutions for the exponent i-1,

there will exist solutions for an exposant not greater than i(i-1).

Some times, we must make attention to the initial change of the data. For example, let the following equationd

 

for some k integers like 7, there are solutions, for others like 2, there is no solution. It is too easy toi pose

 

 

 

 

the lemma 1 is satisfied

 

 

The correct solution is to pose

 

 

 

 

Like this

 

and

 

Conclusion

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The conclusion is that Ghanouchi's sequences, series and numbers have several applications in all diophantine equations, we saw some of them and there are many others like Pilai equation, Smarandache equation, the Catalan equation, etc...