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
U^{n}=X^{n}+Y^{n}
GCD(X,Y) =1
We will consider in this study two equivalent equations. Effectively,
let us pose
u = U^{2n}
x = U^{n}X^{n}
y = U^{n}Y^{n}
z= X^{n}Y^{n}
After a little calculus
And
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
Let us build the sequences. If we pose
x_{1} = x
y_{1} = 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
x_{i}, y_{i} 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
xy = 0, then
. It is confirmed by the fact the limit of the general term of the series (here
xy) 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 U^{c} = X^{a}+Y^{b}; GCD
(X,Y) = 1. If we pose
Then
And
Equation 5 and 6 are the new Fermat equations,
they imply after the same reasoning and formulas than for Fermat equation xy
= U^{c} (X^{a}Y^{b}) = 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(X_{k}) =1
We pose
With k = 1,2,…,i
Then
Equation 7 and 8 are the new Fermat equation,
generalized equation has no solution for n>i (i1) and n_{k}>i (i1)
other than
Then
X = Y = 0
Because GCD (X_{k}) = 1
The new Fermat equations must be used with precaution, for example for
the following equation
kU^{n} = X^{n}+Y^{n}
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), n_{k} ≤ i (i−1), there are solutions,
for example
i = 2 has 3^{2}+4^{2} = 5^{2}
i = 3 has 3^{2}+4^{2}+5^{2} = 6^{3}

95800^{4}+217519^{4}+414560^{4} =
422481^{4} 
i = 4 has 27^{5}+84^{5}+110^{5}+133^{5}
= 144^{5}
It seems to have solutions only for i+1, but we will prove that it is
for i (i1)
We will suppose that X_{k} are coprime, let
Lemma 5
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 (i1). If there are solutions for
the exponent (i1), there will be solutions for the exponent i(i1). It
is not true for i, because of the exponent –(i1)
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
We will build sequences
And
And
And
The expressions are
We prove it by induction, as we did for rational sequences
So
y_{j} 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.