Whole Perfect Vectors and Fermat’s Last Theorem

Whole Perfect Vectors and Fermat’s Last Theorem Ramon Carbó-Dorca Institut de Química Computacional, Universitat de Girona, Girona (Spain) and Ronin Institute, Montclair, NJ (USA) Abstract A naïve 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. Keywords Fermat’s Last Theorem; Whole Perfect Vectors; Sine and Cosine functions; Natural and Rational Vectors; Fermat Vectors 1. Introduction Time has passed since the Wiles 100-page demonstration of Fermat’s last theorem [1]. Meanwhile, our laboratory has been working on several computational aspects of this mentioned theorem. These studies were focused on extending Fermat’s theorem to larger dimensions and involving powers of higher natural numbers [2-5]. 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 selfformulated theorem. The present paper describes a simple proof of Fermat’s last theorem. 2. Whole perfect vectors In three-dimensional vector semispaces, see references [6-10], constructed on the non-negative real set: 𝑉3 (ℝ+ ), one can define a (whole) perfect vector: ⟨𝐩| = (𝑎, 𝑏, 𝑟), see for more information references [10,12], when the vector elements meet the following property: 1 https://doi.org/10.32388/HFXUL0 ⟨𝐩| ∈ 𝑉3 (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: 𝑉3 (ℝ+ ), that is: ∀⟨𝐩| ∈ 𝐏 ⊂ 𝑉3 (ℝ+ ). 3. Minkowski n-th order norms In the perfect vector subset 𝐏, one can define a Minkowski norm of n-th order as follows: ∀⟨𝐩| = (𝑎, 𝑏, 𝑟) ∈ 𝐏: 𝑀𝑛 (⟨𝐩|) = 𝑎𝑛 + 𝑏 𝑛 − 𝑟 𝑛 . (2) Thus, in the subset 𝐏, one can suppose contained a vector set with a BanachMinkowski metric associated with a metric vector: ⟨𝐦| = (1,1, −1). A recent general study of Minkowski metric spaces discussed such vector space structure; references [12-14]. 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 the equation (2). 4. Homothecy and original vector The homotheties of an original perfect vector ⟨𝐩| are defined as: ∀⟨𝐩| ∈ 𝐏 ∧ ∀𝜆 ∈ ℝ+ : ⟨𝐡| = 𝜆⟨𝐩| = (𝜆𝑎, 𝜆𝑏, 𝜆𝑟) ∈ 𝐏. (3) The Minkowski norms of the homotheties ⟨𝐡| of perfect vectors are easily related to the ones associated with a perfect origin vector: 𝑀𝑛 (⟨𝐡|) = (𝜆𝑎)𝑛 + (𝜆𝑏)𝑛 − (𝜆𝑟)𝑛 = 𝜆𝑛 (𝑎𝑛 + 𝑏 𝑛 − 𝑟 𝑛 ) = 𝜆𝑛 𝑀𝑛 (⟨𝐩|). (4) 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. 2 5. Extended Fermat vectors A WP vector can be named as an extended Fermat vector when its secondorder Minkowski norm is null, that is: ∀⟨𝐟| ∈ 𝐅 ⊂ 𝐏: 𝑀2 (⟨𝐟|) = 0 ⇒ 𝑎2 + 𝑏 2 − 𝑟 2 = 0 ↔ 𝑎2 + 𝑏 2 = 𝑟 2 . (5) 6. Natural Fermat vectors 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: 𝑀2 (⟨𝐭|) = 0. (6) The so-called Pythagorean triples are a nickname for true Fermat vectors (of second order). 7. Rational Fermat vectors 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: 𝑀2 (⟨𝐭|) = 0 ⇒ 𝑎 2 𝑏 2 𝑎 𝑏 𝑎2 + 𝑏 2 = 𝑟 2 ⇒ ( ) + ( ) = 1 ⇒ {( ) , ( )} ⊂ ℚ+ . 𝑟 𝑟 𝑟 𝑟 (7) Therefore, the vectors defined over the non-negative rational set have the form: 𝑎 𝑏 ⟨𝐤| = (( ) , ( ) , 1) ∈ 𝐊 → 𝑀2 (⟨𝐤|) = 0, 𝑟 𝑟 (8) and could be considered as extended rational Fermat vectors, with elements constructed over the set ℚ+ , whenever equations (7) and (8) hold. Therefore, by construction, one can write: 3 0<𝑎<𝑏<𝑟↔0< (9) 𝑎 𝑟 𝑏 < < 1, 𝑟 and thus, one can also consider the vectors of the subset 𝐊 as possessing elements defined within the 0,1 unit interval. 8. Isomorphism between natural and rational Fermat vectors 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. (10) One can also symbolically write: 𝑟 −1 (𝐓) = 𝐊 ⇔ 𝑟(𝐊) = 𝐓. (11) Therefore, proving Fermat’s last theorem in the set 𝐓 is the same as proving it in the set 𝐊, and vice versa. 9. Trigonometric Fermat vectors 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 the equation (9) hold. Second, because one initially deals with natural Fermat vectors, one can also write: 𝑎 2 𝑏 2 𝑎2 + 𝑏 2 − 𝑟 2 = 0 ↔ ( ) + ( ) − 1 = 0, 𝑟 𝑟 𝜋 (12) Then, taking angles in the interval 𝛼 ∈ (0, ], due to the symmetrical nature 4 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: 𝜋 ∀𝛼 ∈ (0, ] ∧ 𝐶 = 𝑐𝑜𝑠(𝛼) ; 𝑆 = 𝑠𝑖𝑛(𝛼) : 4 ⟨𝐮| = (𝑆, 𝐶, 1) ∈ 𝐔 → 𝑀2 (⟨𝐮|) = 𝑆 2 + 𝐶 2 − 1 = 0 (13) 4 9.1. Some remarks on trigonometric Fermat vectors Not all the trigonometric vectors ⟨𝐮| ∈ 𝐔 written as in equation (13) 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. (14) 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. 10.Minkowski norms of trigonometric 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: (15) ∀⟨𝐮| = (𝑆, 𝐶, 1) ∈ 𝐔 → ∀𝑛 ∈ ℕ ∧ 𝑛 ≠ 2: 𝑀𝑛 (⟨𝐮|) = 𝑆 𝑛 + 𝐶 𝑛 − 1 ≠ 0 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 [15]. Thus, the formulation will not be explicitly repeated here, except for the 1st, 3rd, and 4th powers, given as a short illustrative example below: 𝑆+𝐶 ≠1 1 𝑆 3 + 𝐶 3 = (3(𝑆 + 𝐶) + 𝑐𝑜𝑠(3𝛼) − 𝑠𝑖𝑛(3𝛼)) ≠ 1 4 1 4 4 𝑆 + 𝐶 = (3 + 𝑐𝑜𝑠(4𝛼)) ≠ 1. (16) 4 Inequality expressions like those shown in the equation (16) 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 (15) inequality. 5 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 𝐓and 𝐊. 11. Discarding the existence of true natural Fermat vectors of order greater than 2 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. To complete Fermat’s theorem proof, a discussion follows of whether one might construct natural (2+1)-dimensional vectors as true natural Fermat vectors of orders higher than the second. 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) ⇒ 𝑀𝑛 (⟨𝐩|) ≠ 0, (17) then one can continue, trying to follow a reductio ad absurdum procedure leading to the demonstration. One can start admitting that the equation (17) 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) ⇒ 𝑀𝑝 (⟨𝐩|) = 0, (18) therefore, if the expression (18) is true, then one can also write: 𝑀𝑝 (⟨𝐩|) = 0 → 𝑎𝑝 + 𝑏 𝑝 − 𝑟 𝑝 = 0 → 𝑎𝑝 + 𝑏 𝑝 = 𝑟 𝑝 𝑎 𝑝 𝑏 𝑝 ⇒( ) +( ) =1 𝑟 𝑟 (19) in the same manner, one can use the above equalities also to write: 𝑎 𝑏 𝑥 = ∧ 𝑦 = → 𝑥 𝑝 + 𝑦 𝑝 = 1. 𝑟 𝑟 (20) 6 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}. (21) Then, owing to the equation (21), one can also write: 𝑎 2 𝑏 2 𝑆 2 + 𝐶 2 = 1 → 𝑥 2 + 𝑦 2 = 1 ⇒ ( ) + ( ) = 1 → 𝑀2 (⟨𝐩|) = 0, 𝑟 𝑟 (22) therefore, the vector ⟨𝐩| of the equation (17) is a rational Fermat vector. Such a result contradicts the existence of Minkowski norms higher than 2, as expressed in the equation (18). Therefore, one cannot obtain natural vectors fulfilling the equation (18) providing Minkowski norms of order larger than two; thus, the equation (17) 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. 12.Discussion 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: 𝑀2 (⟨𝐭|) = 0 → ⟨𝐭| = ⟨𝐟|, (23) 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) ⇒ 𝑀𝑛 (⟨𝐯|) ≠ 0. (24) 7 A result corresponding to how one can formulate Fermat’s last theorem. Acknowledgments The author wishes to acknowledge Blanca Cercas MP 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, U. of Torun. Manuscript comments by Professor Carlos C. Perelman, Ronin Institute, are also greatly acknowledged. References 1. A. Wiles “Modular Elliptic-Curves and Fermat's Last Theorem”. Ann Math 141 (1995) 443-551. 2. R. 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