Extension of Fermat’s last theorem in Minkowski natural spaces (original) (raw)
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Journal of Mathematical Chemistry, 2016
In order to use the structure and operations of Molecular Similarity semispaces, Natural Vector Semispaces are considered in this study as vector spaces defined over the set of natural numbers, with zero added if necessary. The complete sum and inward power of a vector, defined as basic tools in Quantum Molecular Similarity, are now applied to a Natural Vector to describe Minkowski norms in these vector spaces. The structure and behavior of the Minkowski norm of Natural Vector inward powers and the Boolean Hypercube vertex translation into natural numbers are further used to conjecture a plausible general set up of Fermat's Last Theorem.
Refinement of a generalized Fermat’s last theorem conjecture in natural vector spaces
Journal of Mathematical Chemistry, 2017
This study, at the light of a systematic computational search, provides a refined generalized Fermat's last theorem conjecture with respect to the one, which appeared in a recent paper in this journal dealing with a Fermat-like behavior, attached in turn to the Minkowski norms defined within natural vector spaces. The aim of this kind of studies is to connect vector spaces with Boolean Hypercubes. This pretends also uncover the inner structure of vector sets, which appear important for the vectorial description of molecular structures and in this manner lead to new QSPR developments.
Fermat Surfaces and Hypercubes
When observed from a natural vector space viewpoint, Fermat's last theorem appears not as a unique property of natural numbers, but as the bottom line of extended possible issues involving larger dimensions and powers. The fabric of this general Fermat's theorem structure consists of a well-defined set of vectors associated with-N dimensional vector spaces and the Minkowski norms one can define there. Here, this special vector set is studied and named a Fermat surface. The connection between Fermat surfaces and hypercubes is unveiled.
Whole Perfect Vectors and Fermat's Last Theorem
2023
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.
Six Dimension and the Generalized Fermat's Theorem
2023
Fermat's Last Theorem is generalized to Six Dimension by including Temperature as an imaginary component of time and Gravity becomes an imaginary component of energy. Temperature dimension is classical. While the imaginary energy is responsible for gravitons. When time is absent, or is a fixed value, the remaining 5D without time domains represent the enclosing boundary domain to the 6D manifold, with gravity due to the duality between energy and mass, and represent Dark Matter domains. Any mass that might exist inside behaves like Tachyons, with imaginary energy. Due to the imaginary energy, Tachyon-anti-Tachyon if present is in pairs. And they will annihilate each other and produce a static gravity potential.
The geometry of Minkowski spaces — A survey. Part I
Expositiones Mathematicae, 2001
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.
Some works of Furtw\"angler and Vandiver revisited and the Fermat last theorem
arXiv (Cornell University), 2011
From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat ′ s last theorem for p > 3 and to a stronger version called SFLT, by introducing governing fields of the form Q(µ q−1) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2), such that for q | q in Q(µ q−1), we have q 1−c = a p (α) with α ≡ 1 (mod p 2) (where c is the complex conjugation), then Fermat ′ s last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(µ q−1). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researches (probably of analytic or geometric nature). Résumé. Reprenant des travaux de P. Furtwängler et H.S. Vandiver, nous posons les bases d'une nouvelle approche cyclotomique du dernier théorème de Fermat pour p > 3 et d'une version plus forte appelée SFLT, en introduisant des corps gouvernants de la forme Q(µ q−1) pour q premier. Nous prouvons par exemple que s'il existe une infinité de nombres premiers q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2), tels que pour q | q dans Q(µ q−1), on ait q 1−c = a p (α) avec α ≡ 1 (mod p 2) (où c est la conjugaison complexe), alors le théorème de Fermat est vrai pour p. Plus généralement, le but principal de l'article est de montrer que l'existence de solutions non triviales pour SFLT implique de fortes contraintes sur l'arithmétique des corps Q(µ q−1). A partir de là, nous donnons des conditions suffisantes de non existence qui nécessiteraient des investigations supplémentaires pour conduireà une preuve de SFLT, et nous formulons diverses conjectures. Ce texte doitêtre considéré comme un outil de base pour de futures recherches (probablement analytiques ou géométriques). ******* This second version includes some corrections in the English language, an in depth study of the case p = 3 (especially Theorem 8), further details on some conjectures, and some minor mathematical improvements. 1 Equation (u + v ζ) Z[ζ] = w p 1 or p w p 1 , in integers u, v with g.c.d. (u, v) = 1, equivalent to N K/Q (u + v ζ) = w p 1 or p w p 1 , where ζ := e 2iπ/p , K := Q(ζ), p := (ζ − 1) Z[ζ] (see Conjecture 1). Remark that the important condition g.c.d. (u, v) = 1 implies w 1 prime to p. Note that if u v = 0, the condition g.c.d. (u, v) = 1 implies (u, v) = (±1, 0) or (0, ±1). 2 If ν ≥ 1, then α := a+c ζ a+c ζ −1 is a pseudo-unit (i.e., the pth power of an ideal), congruent to 1 modulo p; so, from [Gr1, Theorem 2.2, Remark 2.3, (ii)], α is locally a pth power in K giving easily α ≡ 1 (mod p p+1), then c (ζ−ζ −1) a+c ζ −1 ≡ 0 (mod p p+1), hence c ≡ 0 (mod p 2). 3 If u − v ≡ 0 (mod p), then α := uζ+v u+vζ is a pseudo-unit congruent to 1 modulo p; so, from [Gr1, Theorem 2.2, Remark 2.3, (ii)], α is locally a pth power giving α ≡ 1 (mod p p+1), then (u−v)(ζ−1) u+vζ ≡ 0 (mod p p+1), hence u − v ≡ 0 (mod p 2). This is valid in the Fermat case if x − z ≡ 0 (mod p), and gives x − z ≡ 0 (mod p 2).
Error-correcting codes and Minkowski’s conjecture
Tatra Mountains Mathematical Publications, 2010
The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a2+b2 = c2. Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski’s conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes.
On the construction of N-dimensional hypernumbers
2011
Complex numbers extend the concept of the 1 dimensional numbers to 2 dimensions. Quaternions extend numbers to 4 dimensions. Octonions and sedenions are extensions to 8 and 16 dimensions respectively. We study a general form of complex numbers, various axiomatic structures of 3 dimensional numbers, and finally N dimensional numbers, 2 k N , k=0,1,2,.... Quaternions, octonions and sedenions are special cases. Two different structures for addition are studied.