Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I ()
1. Introduction
1.1. Some Historical Reminders about Fermat’s Theorem
In this paper, we are interested in some works of the french mathematician Pierre de Fermat (1601 - 1665), specially his well-known “solution” or problem of solving the equation: an + bn = cn, problem written by himself in the margin of his edition of Diophantus cf. [1] , some day between 1621 and 1665, and called his Last Theorem, and who became world famous after his death.
From a historical point of view, to solve this problem, Fermat took advantage of, and promoted worldwide the so-called “infinite descent” method, which already existed in [1] , that is a posthumous publication in 1670, 5 years after his death, by his son Samuel de Fermat. This method was originally used by Fermat himself, when solving the 20th Diophantus problem cf [1] , i.e. “In
:
can’t be a square with
” i.e. to show the impossibility of squaring the area of entire right-angled triangles, and which is an effective means of proof. The usual and easy way to solve this problem is the one proposed in [2] , which consist to proove that the equation X4 + Y4 = Z2 obtained from a4 + b4 = c4, is not solvable. The method used, is infinite descent . Following Fermat, many equally famous authors, ranging from Euler 1738 to Carmichael 1913, via Vranceanu 1966 (cf. [3] p. 15), proposed demonstrations all based on finality on this same method of infinite descent (except for certain demonstrations using the ring of integers of Gauss).
But in the end, the proof done in 1994 for the general case ap + bp = cp, by the British mathematician Andrew Wiles cf. [4] , somehow closed the problem of resolution.
This proof is due to the culmination of new methods developed in the 20th century, located at the common frontiers of algebraic number theory, arithmetic, algebraic geometry and complex analysis, focused on the properties of certain types of analytic functions called modular forms, dealing with certain conjectures about elliptic curves and modular forms (these conjectures were all fully proven, in 1986, 1994 and 1999).
It was in 1955 that Tanyama and Shimura announced their conjecture (rediscovered by Weil in 1967): which says that “Any elliptic curve is modular”. In 1984, Hellegouarch and Frey, notice that it was possible to associate an elliptic curve with an eventual solution of Fermat, in this way:
, where
;
with
. This Hellegouarch-Frey elliptic curve with strange discriminant
, is “semi-stable”, and does not seem to be modular; indeed, this last hypothesis was proven by Ribet in 1986 cf. [5] , using his proof for some cases of Serre’s ε-conjecture; And it was finally in 1994 that the Tanyama-Shimura-Weil conjecture was proved by Wiles in 1994 cf. [4] [6] , (for cases of semi-stable elliptic curves) which meant that Fermat’s last theorem was totally proven.
1.2. The Case of Fermat’s Theorem, for n = 4
In this article, we particularly focus on the case n=4, that is to say, in the resolution of the equation a4 + b4 = c4, which as we know, only admits the solutions generated by the trivial solution
, and its associates (cf. Proposition 2.1.).
The question that still arises for solving this equation is the following:
Is there a Diophantine demonstration other than that: complex one? Wiles’ type? or of the classic infinite descent using equation X4 + Y4 = Z2 ? (This last one being closely linked to the resolution of many famous Diophantine equations of order 4, as for example, those related to the old problem of the impossibility of squaring the area of a right angled triangle).
The answer somewhere is yes, like for example in [7] , where one can find infinite descent using only equations like
. But the calculations remain difficult, and no method is extracted from [7] , which could serve as a tool to progress on other cases of Fermat’s equations.
Conversely, one of the main objectives of our article is precisely to propose a standard resolution method, usable for n = 4, but also for n = 2p or p, where p is prime. This method is based on some very particular divisors of a and b.
2. Utilities for the Proof of Fermat’s Theorem for n = 4
2.1. Key Idea of the Proof for the Resolution of a4 + b4 = c4
The goal of our article is to prove Theorem 3.1., i.e. Fermat’s Theorem when n = 4, in the way that we are looking for a practical proof, using arithmetic invariants, that we will define, and known as valuable arithmetic tools, to have been used in our other works.
So we consider in
, an eventual solution:
, pairwise prime,
odd, and
, being minimal in value. (2.1)
From there, as a consequence of the Pythagorician divisors Theorem, cf. Theorem 2.1. & Lemma 2.1., see also [8] , and through our main result on Fermat’s b-absolute divisors, and expressed in Theorem 2.2., it was possible for us to obtain a contradictory result, which is: There is another solution
, pairwise prime,
odd, such that:
; But this is impossible, because of the minimality condition of s, defined in (2.1).
Furthermore, we note that: In the case
, we have:
, which necessarily implies that
, cf. Proposition 2.3.(i), which is impossible.
Roadmap and Articulations of the Proof
How to prove Theorem 3.1.? Consider a solution
of equation (2.1.), and take a look at the even term
, then necessarily
and
are
odd, and therefore we can define
such that
; as well as the quantity
.
Taking into account the Fermat’s equation
as a Pythagoras equation, and applying the Pythagorician divisors Theorem, cf. Theorem 2.1. & Lemma 2.1.:
It has been possible to find 2 proper divisors
and
of
, defined by:
and
, where
.
And finally to determine a third odd proper divisor
of
by the equality
.
In particular
, where
are odd and pairwise prime.
And in addition:
We call Fermat’s absolute b-divisor, the following pairs of integers:
.
These very particular divisors of the even term
, verify the following Pythagorean equation:
The central element being even, the others odd.
From this last Pythagorean equation, it suffices to use a parameterization for Pythagorician triplet, of the type:
, with
,
, and
, then we deduce the existence of integers
, and coprime, such that
, which contradicts the minimality of s.
2.2. Notations and Reminders for Pythagoras and Fermat Equations
Let’s consider:
a2 + b2 = c2, the Pythagoras equation. (2.2)
And:
a4 + b4 = c4, the Fermat’s equation. (2.3)
Remark 2.1 For equation (2.3) we could suppose that positive trivial solution exist, for at the end, proove the impossibility, of such solution.
Then cf. [8] and [9] we have:
Proposition 2.1 The set of all the solutions of the Pythagoras equation a2 + b2 = c2 (resp. and eventually these of the Fermat’s equation: a4 + b4 = c4), is formed from the solutions generated by all the positive primitive solutions, and their associates.
Remark 2.2 1) Let
checking (2.2), then
.
2) Let
checking (2.3), then
, with
,
, and
where p is an odd prime and
. In particular
with
and necessarily
odd. (see Proposition 2.3. for the proof).
This in no way restricts the expression of the generality of the solutions of said equations, because
is also a solution called “associated with
”, such that
.
Definition 2.1 We now define the following sets cf. [8] :
1)
: The set of non-trivial, primitive and positive Pythagorician solutions of the type
. This set is exactely the set of non-trivial, primitive and positive Pythagorician solutions.
2)
: The set of Fermat equation’ solutions
, which are non-trivial, primitive and positives of type
, with:
,
, and
where p is any odd prime with
. This set is exactely the set of non-trivial, primitive and positive solutions of Fermat equation’ solutions. cf. Proposition 2.3.
Let us recall cf. [3] [10] , that (see also [8] for another important parametrization):
Proposition 2.2
.
2.3. Pythagorician Divisors Theorem Applied to Fermat’s Equation a4 + b4 = c4
2.3.1. Pythagorician Divisors
Let’s remind the Pythagorician divisors Theorem cf. [8] :
Theorem 2.1 Let
,
,
odd,
pairwise prime, then there are equivalences between the following propositions:
(1)
; (ii)
; (iii)
.
Where in this Theorem, the notations are:
Definition 2.2 1)
is defined by:
.
2)
.
3) The pythagorician divisors
(resp.
) are defined by:
Remark 2.3 As results: 1)
,
;
2)
;
and
.
Remark 2.4 Cf. [8] , we get the important relations:
and
(2.4)
2.3.2. Some Important Results on Fermat’s Equation a4 + b4 = c4
Let’s now particularly consider the Fermat equation:
,
; with goal to conclude that
; with
,
, where
odd, and
(cf. Proposition 2.3.(i)) put for the 2-adic valuation of b, and choosed among all solutions of (2.3) such that s ≥ 2 have minimal value; moreover a, b1 and c are odd.
We have the following Proposition.
Proposition 2.3 Let
, relatively pairwise prime/
, then
cf. Definition 2.1.(2), that means that:
i)
, in particular
.
ii)
(i.e.
), and
where p is an odd prime.
iii)
.
Proof 1 (i) Let
took as in Lemma
. Let’s put
, then:
(cf. Proposition 2.2)
,
such that
and in addition
, while
because a is odd.
In conclusion:
.
(ii)
Suppose that
and
i.e.
.
Let
. In this case, considering the Pythagoras equation:
, it comes:
, i.e
cf. Definition
2.2. (1). So according to [8] or Remark 2.3.(2) gives:
, therefore, since
and that e is even, then
is odd; but then
.
But Theorem 2.1 gives:
, which is absurd because
. So as stated:
,
.
Similarly, for the other result: suppose that
,
, that is to say a power of an odd natural prime number p.
Let us put:
according to Definition 2.2., then we have:
and from Proposition 2.4. (3), saying
.
We deduce that: either
; either
.
Now the Theorem 2.1.
- In the first case:
, which is absdurd.
- In the second case:
and
; which is absurd.
Thus:
,
where p prime.
Conclusion: Neither a nor b can be a power of a prime natural number.
(iii)
Suppose that
.
Then
,
such that
and
, but then:
, that is impossible, since the difference of two squares cannot be equal to 1.
Suppose that
. If
, that is absurd; If
that is absurd.
As a consequence as claimed
.
See also an altenative proof in [9] pages 59-60-61.
Remark 2.5 Subsequently, as already said, when
, i.e.
, then we have:
;
and
,
,
, an odd prime.
2.3.3. Direct Application of the Pythagorician Divisors Theorem to Fermat’s Equation: a4 + b4 = c4
We now directly apply the results of the Pythagorician divisors Theorem to Fermat’s equation:
, because
:
Proposition 2.4 Consider Fermat’s equation:
, and let
, a non-trivial hypothetical solution, of it.
Then:
/
and
/
, such that:
(i)
and
.
(ii)
(2.5)
Proof 2 Indeed Pythagorician triplet:
is a non-trivial,
primitive and positive one. Its Pythagorician divisors are in this case (since
, i.e
):
because remark 2.3 (2)
(2.6)
The application of Theorem 2.1. where in addition one takes into account the results of Proposition 2.4, gives the following corollary (the notations remaining unchanged).
Lemma 2.1 Consider
a positive triplet. There are equivalences between the following 3 propositions:
(i)
.
(ii)
.
(iii)
.
Remark 2.6 1) Note that:
,
.
2) Note that necessarily:
and
(otherwise
would be trivial, which would be contradictory).
Proof 3 It suffices to apply Theorem 2.1 to the Pythagorician triplet
taking into account the formulas (2.5).
But, let us continue as we are looking through Theorem 2.2., for a new proof of our proof.
We give the following proposition, which we will need in the next paragraph:
Proposition 2.5 The notations being the same as those of Lemma 2.1., we have:
.
Proof 4 From Remark 2.6.:
.
For proving
: Lemma 2.1. (ii) & (iii) implies that:
And:
But then :
; from where:
.
2.3.4. Other Classical Demonstrations, Using Well-Known Diophantine Equations of Degree 4
Remark 2.7 Taking account Remarks 2.4. & 2.6., as well as Propositions 2.3.. & 2.5, it is easy to give all the list of all the famous well-known diophantine equations of degree four, connected with the impossible resolution of
, equation whose these coefficients would be solutions; however, all of them admit only trivial solutions (or other are impossible), and if solution are not trivials, this would imply that
or
, which would be absurd.
All this would allow us to conclude right now, that
.
Remark 2.8 As an exemple, because it will be to long to expose all this equations, we give the following: Applying Remark 2.4.(i) to
, in relation to Proposition 2.4. above, it comes that Pythagorician triplet
, is such that (cf. Remark 2.4.):
and
, which implies that:
,
But this means that the area of this right-angled triangle is the square:
;
Wich is absurd, because that is the famous impossible squaring of the right-angled triangle area problem or 20th Diophantus Problem, proved by Fermat’s result (cf. also [11] for a new proof).
This equation which is also (because
) of the type:
(i.e. an Euler equation) and does not admit solutions in integers, such that all of which are different from zero (cf. [12] p. 70, Exercice 1 or else [13] p. 38).
2.4. Concept of Absolute Fermat Divisors
We are now going to specify what are the integers
and
, which appeared naturally, during factorizations. For that, we will introduce the notion of: “Absolute Fermat’s divisors”.
Consider first the following lemma:
Lemma 2.2 let
, a non-trivial hypothetical solution of Fermat’s equation:
. Recall that
, and that
.
Let’s set
and
, then:
divides
and
divides
, and so we can set:
and
.
Proof 5 We use parts of proof of Theorem 2.2. hereafter, and the fact that
is algebraically closed. Indeed:
is necessarily an integer because (cf. proof of Theorem 2.2.):
and
is a rational root of
.
Similarly
is an integer too (i.e.
) because:
Let
such that
, this implies that
, consequently
, but ( cf. proof of Theorem 2.2) hereafter,
is a rational root of
.
This allows us to define the absolute Fermat divisors.
2.4.1. Definition of Absolute Fermat Divisors
Definition 2.3 Consider Fermat’s equation:
, and let
, be a not trivial eventual solution of it. We call Fermat’s absolute a-divisor (resp. Fermat’s absolute b-divisor) the following pairs of
integers:
(resp.
.
is said to be the co-adjunct divisor of
and vice versa. We have the same notion for
with respect to
.
Definition 2.4 Let
, then we set
such that
, and
. Note that
.
Proposition 2.6 Let
. Then:
1) There exists
with
, such that:
and
. Then:
and
and
.
2)
(cf. formula (2.5))/
.
In addition
; and
if
, while
;
if
. So
(resp.
) is a proper divisor of
.
Proof 6 See parts of proof of Theorem 2.2., in particular points around formula (2.7), and also (2.5) and(2.6).
Remark 2.9 Then we’ll set
and so
cf. Definition 2.3. Then (see the proof of Theorem 2.2.):
and
, where
cf. Remark 2.6.(1).
2.4.2. Fermat’s Absolute Divisors Theorem
The notations being unchanged, we get “Fermat’s absolute divisors Theorem”:
Theorem 2.2 ((First form) of Fermat’s absolute divisors).
There is equivalence between the following propositions:
(i)
i.e.
is realized.
(ii)
, coprime in pairs, for wich:
coprime in pairs, and
, checking
, and such that for notations:
, with
defined by
,
and
, where
, then the following system is checked:
Remark 2.10 Note that equations (i) and (ii) are equivalent to (iii).
(iii)
odds, coprime in pairs, such that:
and:
.
And that in this case:
and
otherwise
and
, which would be absurd.
As a result
are proper divisors of
.
Remark 2.11 Note that cf. Remark 2.6. and Propositions 2.3. & 2.5, we get:
when
, (resp.
when
) and they are all odds, relatively pairwise prime, and are respectively proper divisors of
, since we have the factorizations:
; as
.
Reminders 2.1
,
,
, consider
, where
. Then:
.
In particular
.
Let’s now show this theorem.
Proof 7
The equivalences between the systems, within points ii) and iii) are obvious.
Moreover, it is clear that (ii)
(i) and (iii)
(i).
It therefore remains to show that: (i)
(ii) (and that (i)
(iii) What we will not do here, for the sake of simplifying the results).
Note that from:
, we get:
- Let us show that i)Þii). We know that
.
1) Case 1:
i.e
.
.
From where:
(2.7)
But:
and
and
even,
odd because
and
.
But then:
with
,
but:
, therefore:
. We therefore have:
. Let’s put
.
Consequently
and
odds such that:
(2.8)
Checking:
Point (iii) of the theorem, when
.
As a result in this case:
((2.7)
)
(2.9)
2) Case 2:
i.e
.
From where:
(2.10)
But:
and
.
Which implies:
and
,
with
odd and
even, because
is even.
Then:
. And so:
.
Consequently:
.
From where:
such that
, (here
) moreover, considering
, it comes:
We then set:
which is quite an integer (cf. Lemma 2.1.).
Then:
.
In the end, we have the formulas of the Theorem 2.2.(iii) when
:
Summaring, and considering (2.10), the quantities
verify:
.
(11)
3. New Proof of Fermat’s Theorem for n = 4
We get the following corollary:
Corollary 3.1 Let
, and
defined by:
.
Then we have the proposition:
, formed by strictly odd divisors of
, checking
, and such that the following equation of degree 8 holds:
, where:
and
.
And:
The proof is obvious, it suffice to calculate
, from the values c and a extrated from Theorem 2.2.
Remark 3.1 1) The degree 8 equation:
, is of the type of that of [7] , but ours is more precise, both at the level of the exponent “s”, which is the key to our proof, as well as the other quantities which are in fact proper divisors of the odd part of b, and whose product gives back this same odd part.
2) At this stage, we could also conclude in a classic way, because we find a right triangle whose area is equal to the square
which is impossible.
From this we deduce the goal of our paper:
Theorem 3.1 Fermat’s equation:
, does not admit solutions in
(i.e.
). Consequently its only solutions are those generated by the trivial one
and its associates.
Remark 3.2 From Theorem 2.2. and Corollary 3.1., other new and original proofs exist, but the following giving is the most natural, and uses the argument of the infinite descent or the minimality of a parameter, between 2 similar Fermat equations.
Proof 8 Consider
, with
and minimal in all solutions of
.
Clearly
,
,
,
, and
, checking:
, and coprime, such that:
, and coprime such that
, and
.
But these last 2 Fermat’s equations, contradict the minimality of s: that is absurd.
Remark 3.3 About the proof of Theorem 3.1.
An alternative proof, which is a very particular recurrence, is to consider for fixed k,
, the following property:
: “The equation
, with
odds; is not solvable”.
It is clear that
is true and that
is true (cf. Proposition 2.3.(i)).
Let’s make the following recurrence hypothesis:
“
is true from rank 0, up to rank
”.
Let us deduce that
is true for
, which means that
would be true
.
Indeed, suppose the converse when
, i.e.
is false. Then:
such that
.
But then, the previous demonstration shows precisely the existence of
such that:
which means that
is false, thus contradicting the recurrence hypothesis, that is absurd.
Remark 3.4 1) Always about proof of Theorem 3.1.
An other alternative proof, is to consider a solution:
, i.e.
with
arbitrary (i.e. s not necessarily minimal) then after
iterations as was done once in proof 8, we will have an equation of type
, in odd integers, that is not solvable cf. Proposition 2.3.(i).
2) And what about the remaining, 8 degree equation coming from quantity a:
? that extracted from the system of Remark 2.10., which follows Theorem 2.2, and which can be reduced to a Pythagorean equation or to an equation of the type [14] , see also [8] . We do believe that a priori, it would also be able to produce an original and simple solution.
4. Conclusions and Perspectives
The use of Pythagorician divisors and Fermat’s absolute divisors, will have allowed us to establish a simple and new method based on particular divisors of b, here exposed, for demonstrate the Fermat’s great Theorem:
when
.
We do believe that this method can be used to bring something new in the Diophantine proof of Fermat’s Theorem, when
, i.e.
, p prime, cf. [15] , and finally to the general case of Fermat’s Theorem
, cf. [16] [17] and that, whether p/abc or not.