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An Algebraic-Analytic Approach of Diophantine Equations



Jamel Ghanouchi
 
ABSTRACT

Our purpose in this research is to show how much Fermat equation is rich in analytic applications. Effectively, this equation allows to build amazing sequences, series and numbers. The question of the elementary proof of the theorem remains of course, we will see it in this communication. We will make also an allusion tothe very known Fermat numbers We will see how this problem of the proof is actual and how it can be solved using Fermat sequences and series.

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  How to cite this article:

Jamel Ghanouchi , 2008. An Algebraic-Analytic Approach of Diophantine Equations. Asian Journal of Algebra, 1: 15-24.

DOI: 10.3923/aja.2008.15.24

URL: https://scialert.net/abstract/?doi=aja.2008.15.24

INTRODUCTION

We show that Fermat equation allows to build rational sequences and series. After the formulation of those sequences and series, we calculate their limits. We generalize the sequences and series and their limits to Beal equation and to a generalized Diophantine equation. We define also complex sequences. Of course, all the development is available for other Diophantine equations, we show an example, but there are many others, like Pilai, Catalan and Smarandache equations.

THE SEQUENCES

Fermat equation is

Un=Xn+Yn
GCD(X,Y) =1

We will consider in this study two equivalent equations. Effectively, let us pose

u = U2n
x = UnXn
y = UnYn
z= XnYn

After a little calculus

(1)

And

(2)

We deduce that if U, X, Y, are integers verifying Fermat equation, then u, x, y, z as defined verify simultaneously the new Fermat Eq. 1 and 2 which follow:

Lemma 1

u=x+y
(3)

(4)

Let us build the sequences. If we pose

x1 = x
y1 = y

And integers verifying

And

Then

And

We pose

Also

We pose

And

Which means that

and


Because verifying

The process is available until infinity. For i

And of course

We have built the sequences.


Lemma 2

xi, yi have an expression
(H)
(H`)

Proof of Lemma 2

By traditional induction, for i = 2

Also

We suppose (H) and (H`) true for i, then

Also

It is proved, but

Then, for x ≠ y

Lemma 4

Fermat equation has the constant

THE SERIES

As we saw

It implies the following sum

then

And the limits, if x>y

And

If x < y


Let us study series.

And if x < y then

The Applications of the Sequences and the Series

We will consider firstly that x>y

Also

Then

And

We will study now the convergence of the series. As is convergent and and

Is convergent. It implies that is convergent. Then

It means one thing: and x-y = 0, then . It is confirmed by the fact the limit of the general term of the series (here x-y) is equal to zero, because is convergent. And and X = Y = 0, because

GCD(X,Y)=1 (The reasoning is the same for x < y)

Our question is now: Why are there solutions for n = 2? The answer is in the equations. Effectively, there are trivially an infinity of solutions for n = 1. But the sequences for n = 1 are as it follows;

these are the expressions of the sequences and they do not guarantee the existence of the series for i = 2. So, the case n = 2 is the only exception.

Other Applications of the Sequences and the Series

Let the Beal equation Uc = Xa+Yb; GCD (X,Y) = 1. If we pose

Then

u=x+y
(5)

And

(6)

Equation 5 and 6 are the new Fermat equations, they imply after the same reasoning and formulas than for Fermat equation x-y = Uc (Xa-Yb) = 0

Which means

X = Y = 0

Because GCD (X,Y) = 1

Then Beal equation has not solutions, with the same calculus and reasoning than for Fermat equation, for c>2 and a>2 and b>2.

Now, let the general following equation


GCD(Xk) =1

We pose

With k = 1,2,…,i

Then

u = x+y
(7)

(8)

Equation 7 and 8 are the new Fermat equation, generalized equation has no solution for n>i (i-1) and nk>i (i-1) other than

Then

X = Y = 0

Because GCD (Xk) = 1

The new Fermat equations must be used with precaution, for example for the following equation

kUn = Xn+Yn

There are solutions for k = 7 and there are not for k = 2. We must pose judiciously

And the new equations are

Which are not new Fermat equations and have not the same solutions. It is false to pose

Conclusion

The new Fermat equations allow to build sequences and series which allows to test the impossibility of the resolution of an equation. If they are a consequence of some Diophantine equations, they remain an intellectual building. They must be used with precaution, but they are very efficient.

Generalization

Now, we will generalize the results. Let the following equation

(E)

We will prove that this equation has not solution for

When n ≤ i (i−1), nk ≤ i (i−1), there are solutions, for example

i = 2 has 32+42 = 52

i = 3 has 32+42+52 = 63

  958004+2175194+4145604 = 4224814

i = 4 has 275+845+1105+1335 = 1445

It seems to have solutions only for i+1, but we will prove that it is for i (i-1)

We will suppose that Xk are coprime, let

Lemma 5

(9)
(10)

We will define the sequences

Which implies

The reasoning is available until infinity. Then

Lemma 6

(P) is the following expression

Proof of Lemma 6

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 7

The equation (E) conducts to an impossibility, effectively, if we pose

u, x, y and z verify the lemma 1

Which conducts, we saw it, to

Because they are coprime. Now, the question is: why are there solutions for

Let us pose

The expression (P) becomes

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).

Conclusion

The sequences and the series as we defined them have several applications in several diophantine equations, we saw Fermat and Beal, we saw the generalized equation (E), but there are many others like Pilai, Smarandache, Catalan… They are truly very amazing !

THE ALGEBRAIC APPROACH

Now, let Fermat equation

We pose

(11)
(12)

We will build sequences

And

And

And

The expressions are

We prove it by induction, as we did for rational sequences

So

yj is solution of

And

It means that

Also for x'j

So the only solution is

x = y = 0

CONCLUSION

It appeared since the beginning, before the change of the data, that the equation contains a symmetry between x and y. Effectively, we found u = x+y. We broke the symmetry by changing the equation in two equations u = x+y and We have solved the equation and found a method of resolution of Beal equation The conclusion is that Fermat equation (E) conducts always to an impossibility. It is also the case of Beal equation and generalized Fermat equation.

REFERENCES
Ribenboim, P., 1979. 13 Lectures on Fermat's Last Theorem. Springer-Verlag.

Simon, S., 1997. Fermat's Last Theorem, The story of a riddle that confounded the world's great minds for 358 years. Fourth Estate Limited, London.

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