A na ïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.
Time has passed since the Wiles 100-page demonstration of Fermat’s last theorem [
Still, as far as the present author knows, it seems that no new alternative proofs of the theorem exist, conforming to the handwritten note left by Fermat about a supposedly straightforward proof of the famous initial self-formulated theorem, except for other points of view recently published [
The present paper describes a simple proof of Fermat’s last theorem.
In three-dimensional vector semispaces, see References [
〈 p | ∈ V 3 ( ℝ + ) ∧ { a , b , r } ⊂ ℝ + ∧ 0 < a < b < r . (1)
In such a semispace, the perfect vectors can be collected into a subset P of the space: V 3 ( ℝ + ) , that is: ∀ 〈 p | ∈ P ⊂ V 3 ( ℝ + ) .
In the perfect vector subset P , one can define a Minkowski norm of n-th order as follows:
∀ 〈 p | = ( a , b , r ) ∈ P : M n ( 〈 p | ) = a n + b n − r n . (2)
Thus, in the subset P , one can suppose contained a vector set with a Banach-Minkowski metric associated with a metric vector: 〈 m | = ( 1 , 1 , − 1 ) .
A recent general study of Minkowski metric spaces discussed such vector space structure; see References [
For the sake of coherence, a 3-dimensional space with a defined Minkowski norm can be named as (2 + 1)-dimensional, for example: V ( 2 + 1 ) ( ℕ ) , notes a natural semispace where one has defined a Minkowski metric, like in Equation (2).
First-Order Minkowski Norms and Fermat’s Last TheoremBefore proceeding to the interesting body of the Fermat theorem, it might be worthwhile to discuss the trivial case of Fermat’s theorem within the unit power of natural (2 + 1)-dimensional WP vectors. One can write within this restricted first-order situation that:
∀ 〈 p | = ( a , b , r ) ∈ P : M 1 ( 〈 p | ) = a + b − r . (3)
This result means that if one seeks a zero norm, necessarily one has to write:
∀ 〈 f 1 | = ( a , b , a + b ) ∈ P : M 1 ( 〈 f 1 | ) = 0 , (4)
therefore, one could locate the first-order Fermat vectors into the infinite symmetric square matrix defined as:
S = { s I J = I + J | ∀ I , J ∈ ℕ } ⇒ 〈 f 1 | = ( I , J , s I J ) ← ∀ I , J ∈ ℕ ∧ I < J , (5)
that contains infinite first-order Fermat natural vectors, but evidences that not all natural WP vectors are first-order Fermat vectors, because there might be some natural vector with r ≠ s I J .
Also, for the first-order Fermat vectors, one can write the rational formalism that can be deduced from the structure of the first-order vectors and the Minkowski norm, as shown in Equation (4):
∀ 〈 f 1 | = ( a , b , a + b ) ∈ P : M 1 ( 〈 f 1 | ) = 0 ⇒ a a + b + b a + b = 1 , (6)
Calling the two terms in the sum (6): { p a , p b } ⊂ [ 0 , 1 ] , and realizing that they belong to the unit interval is the same as considering the first-order Fermat vectors isomorphic to the set of two-dimensional probability distributions.
The homotheties of an original perfect vector 〈 p | are defined as:
∀ 〈 p | ∈ P ∧ ∀ λ ∈ ℝ + : 〈 h | = λ 〈 p | = ( λ a , λ b , λ r ) ∈ P . (7)
The Minkowski norms of the homotheties 〈 h | of perfect vectors are easily related to the ones associated with a perfect origin vector:
M n ( 〈 h | ) = ( λ a ) n + ( λ b ) n − ( λ r ) n = λ n ( a n + b n − r n ) = λ n M n ( 〈 p | ) . (8)
This above equation corresponds to the fact that a vector with a Minkowski norm-specific value has the homothetic vector norms as the original vector one multiplied by a factor equivalent to the homothecy parameter, powered to the order of the norm.
A WP vector can be named as an extended Fermat vector when its second-order Minkowski norm is null, that is:
∀ 〈 f | ∈ F ⊂ P : M 2 ( 〈 f | ) = 0 ⇒ a 2 + b 2 − r 2 = 0 ↔ a 2 + b 2 = r 2 . (9)
As natural numbers are a subset of non-negative real numbers: ℕ ⊂ ℝ + , there can exist within the extended Fermat vectors subset, some natural Fermat vectors with elements belonging entirely to the natural number set. If this is the case, they can be called shortly true natural Fermat vectors (of second order or order 2) and symbolize their subset with T , that is:
∀ 〈 t | = ( a , b , r ) ∈ T ⊂ F ∧ { a , b , r } ⊂ ℕ : M 2 ( 〈 t | ) = 0 . (10)
The so-called Pythagorean triples are a nickname for true Fermat vectors (of second order).
True Fermat Vectors Construction and First Order Minkowski NormsOne can use a similar argument to the one employed in the subsection of Section 3 to obtain a construct of all the possible second order true natural Fermat vectors, but using the first order Minkowski norm.
Let’s construct first the set of squares of all the natural numbers: ℕ [ 2 ] = { 1 , 2 2 , 3 2 , ⋯ , I 2 , ⋯ } ⊂ ℕ .
Then, one can use the symbols: ∀ z I ∈ ℕ [ 2 ] → z I = I 2 , for the elements of the natural squared numbers. Furthermore, one can construct the infinite-dimensional matrix:
Z = { z I J = z I + z J = I 2 + J 2 | I , J ∈ ℕ ∧ I < J } , (11)
in this manner, one can see one can construct the true natural Fermat vectors, using the upper triangle of the matrix Z .
To prove this claim, first one can form the vectors 〈 f | = ( z I , z J , z I J ) ; second, one can pick up only the elements of the matrix Z belonging to the set of squared natural numbers, then one can write:
∀ I < J ∧ z I J ∈ ℕ [ 2 ] ∧ ∀ 〈 f | = ( z I , z J , z I J ) ∈ V ( 2 + 1 ) ( ℕ ) → M 1 ( 〈 f | ) = z I + z J − z I J = 0 (12)
which can be seen as an algorithm to construct the true Fermat vectors of second order.
On the other hand, any true Fermat vector can be transformed into a vector with elements defined within the non-negative rational number set: ℚ + ⊂ ℝ + . Such a possibility is easy to consider, as it can be written:
∀ 〈 t | = ( a , b , r ) ∈ T : M 2 ( 〈 t | ) = 0 ⇒ a 2 + b 2 = r 2 ⇒ ( a r ) 2 + ( b r ) 2 = 1 ⇒ { ( a r ) , ( b r ) } ⊂ ℚ + . (13)
Therefore, the vectors defined over the non-negative rational set have the form:
〈 k | = ( ( a r ) , ( b r ) , 1 ) ∈ K → M 2 ( 〈 k | ) = 0 , (14)
and could be considered as extended rational Fermat vectors, with elements constructed over the set ℚ + , whenever Equations (13) and (14) hold.
Therefore, by construction, one can write:
0 < a < b < r ↔ 0 < a r < b r < 1 , (15)
and thus, one can also consider the vectors of the subset K as possessing elements defined within the ( 0 , 1 ] unit interval.
In fact, true natural Fermat vectors and rational Fermat vector sets are isomorphic via a homothecy, which can also be accepted as acting like an operator, that is:
∀ 〈 t | ∈ T : ∃ r − 1 〈 t | = 〈 k | ∈ K ⇔ ∀ 〈 k | ∈ K : ∃ r 〈 k | = 〈 t | ∈ T . (16)
One can also symbolically write:
r − 1 ( T ) = K ⇔ r ( K ) = T . (17)
Therefore, proving Fermat’s last theorem in the set T is the same as proving it in the set K , and vice versa.
Such an isomorphism between true natural and rational Fermat vectors is essential because the vectors in K can be rewritten with trigonometric functions.
First, note that as the true Fermat vectors are whole perfect vectors, one can suppose that the relations of Equation (15) hold. Second, because one initially deals with natural Fermat vectors, one can also write:
a 2 + b 2 − r 2 = 0 ↔ ( a r ) 2 + ( b r ) 2 − 1 = 0 , (18)
Then, taking angles in the interval α ∈ ( 0 , π 4 ] , due to the symmetrical nature of the sine and cosine functions, one can write the true natural Fermat vectors as vectors possessing trigonometric functions as elements, instead of divisions of two natural numbers, that is:
∀ α ∈ ( 0 , π 4 ] ∧ C = cos ( α ) ; S = sin ( α ) : 〈 u | = ( S , C , 1 ) ∈ U → M 2 ( 〈 u | ) = S 2 + C 2 − 1 = 0 (19)
Some Remarks on Trigonometric Fermat VectorsNot all the trigonometric vectors 〈 u | ∈ U written as in Equation (19) could be associated to the rational Fermat vectors 〈 k | .
Trigonometric vectors of type 〈 u | can be seen as extended Fermat vectors. But because all true natural Fermat vectors 〈 t | can generate rational Fermat vectors 〈 k | , one can undoubtedly write that:
K ⊂ U ⇒ r − 1 ( T ) ∈ U → r ( K ) = T . (20)
As we have seen, one can transform all true natural Fermat vectors into rational Fermat vectors, which can also be expressed as trigonometric Fermat vectors.
Then, due to the nature of the expressions of the powers of the sine and cosine functions, and in compliance with Fermat’s theorem, one can write that:
∀ 〈 u | = ( S , C , 1 ) ∈ U → ∀ n ∈ ℕ ∧ n ≠ 2 : M n ( 〈 u | ) = S n + C n − 1 ≠ 0 (21)
Such inequality can be easily proven upon knowing the sum of natural powers expressions of both sine and cosine functions. The most revealing source can be found in Reference [
S + C ≠ 1 S 3 + C 3 = 1 4 ( 3 ( S + C ) + cos ( 3 α ) − sin ( 3 α ) ) ≠ 1 S 4 + C 4 = 1 4 ( 3 + cos ( 4 α ) ) ≠ 1 (22)
Inequality expressions like those shown in Equation (22) can be easily seen as different from unity in general. Sums of larger powers are readily available, yielding terms for the sums of powers, which also clearly differ from the unity.
Therefore, the natural true natural Fermat vectors are isomorphic to some trigonometric vectors 〈 u | , and their Minkowski norms satisfy the above Equation (21) inequality.
Hence, a Fermat last theorem holds for all rational Fermat vectors and thus has to hold for the true natural Fermat vectors because of the isomorphism early discussed between the sets T and K .
In previous sections, one has implicitly shown Fermat’s last theorem. One can suppose such a demonstration included within the definition of true natural Fermat (2 + 1)-dimensional vectors by an attached Minkowski norm. This fact makes the existence of true natural Fermat vectors of order higher than two impossible.
A discussion follows of whether one might construct natural (2 + 1)-dimensional vectors as true natural Fermat vectors of orders higher than the second, as describing a complete Fermat’s theorem proof.
Suppose one wants to demonstrate that vectors in any (2 + 1)-dimensional natural vector space with a well-defined Minkowski norm cannot be true natural Fermat vectors of order higher than 2.
That is:
∀ n ∈ ℕ ∧ n > 2 : ∀ 〈 p | = ( a , b , r ) ∈ V ( 2 + 1 ) ( ℕ ) ⇒ M n ( 〈 p | ) ≠ 0 , (23)
then one can continue, trying to follow a reductio ad absurdum procedure leading to the demonstration.
One can start admitting that Equation (23) is false, so one can write the following property for some natural vector and Minkowski norm order:
∃ p ∈ ℕ ∧ p > 2 : ∃ 〈 p | = ( a , b , r ) ∈ V ( 2 + 1 ) ( ℕ ) ⇒ M p ( 〈 p | ) = 0 , (24)
therefore, if the expression (24) is true, then one can also write:
M p ( 〈 p | ) = 0 → a p + b p − r p = 0 → a p + b p = r p ⇒ ( a r ) p + ( b r ) p = 1 (25)
in the same manner, one can use the above equalities also to write:
x = a r ∧ y = b r → x p + y p = 1 . (26)
However, the pair of rational numbers: { x , y } ⊂ ℚ + , corresponds to the Cartesian coordinates of a point situated into a circumference of unit radius.
One can admit such a previous affirmation following a similar reasoning as in Section 9. Choosing the appropriate angle in trigonometric coordinates, one can write:
{ x , y } → { sin ( α ) , cos ( α ) } ≡ { S , C } . (27)
Then, owing to Equation (27), one can also write:
S 2 + C 2 = 1 → x 2 + y 2 = 1 ⇒ ( a r ) 2 + ( b r ) 2 = 1 → M 2 ( 〈 p | ) = 0 , (28)
therefore, the vector 〈 p | of Equation (23) is a rational Fermat vector. Such a result contradicts the existence of Minkowski norms higher than 2, as expressed in Equation (24).
Therefore, one cannot obtain natural vectors fulfilling Equation (24) providing Minkowski norms of order larger than two; thus, Equation (23) must be true.
As a result, one can say that true natural Fermat vectors of order higher than two cannot exist.
Moreover, this implies that in the context of the present study, only true natural Fermat vectors of dimension (2 + 1) and order two are relevant.
One can write that some whole perfect natural vectors fulfill the equation concerning the nullity of the second-order Minkowski norm:
∃ 〈 t | = ( a , b , r ) ∈ T : M 2 ( 〈 t | ) = 0 → 〈 t | = 〈 f | , (29)
defining in this way true Fermat vectors of second order.
Here, in general, one has deduced, via the isomorphism between natural and rational Fermat vectors, under a trigonometric representation, that true natural Fermat vectors cannot possess Minkowski null norms other than the second-order ones:
∀ n ∈ ℕ ∧ n > 2 ∧ ∀ 〈 v | = ( a , b , r ) ∈ V ( 2 + 1 ) ( ℕ ) ⇒ M n ( 〈 v | ) ≠ 0 . (30)
A result corresponding to how one can reformulate Fermat’s Last Theorem.
The author wishes to acknowledge Blanca Cercas M. P. for her continued support. Without her constant care, this paper could never have been written. The author has also enjoyed an enlightening discussion about the present paper’s form with Professor Jacek Karwowski, University of Torun. Manuscript comments by Professor Carlos C. Perelman, Ronin Institute, are also greatly acknowledged.
The author declares no conflicts of interest regarding the publication of this paper.
Carbó-Dorca, R. (2024) Whole Perfect Vectors and Fermat’s Last Theorem. Journal of Applied Mathematics and Physics, 12, 34-42. https://doi.org/10.4236/jamp.2024.121004