Fermat surfaces and hypercubes

Ramon Carbó-Dorca, Debraj Nath

Article ID: 2490
Vol 2, Issue 1, 2024
DOI: https://doi.org/10.54517/mss.v2i1.2490
Received: 12 January 2024; Accepted: 18 February 2024; Available online: 27 February 2024;
Issue release: 30 June 2024

VIEWS - 699 (Abstract)

Download PDF

Abstract

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 when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.


Keywords

Fermat surfaces; Fermat last theorem; whole vectors; perfect vectors; vector semispaces; Fermat vectors; unit shell; Fermat extended theorem; natural vector spaces; Minkowski norms


References

1. Wiles A. Modular Elliptic Curves and Fermat’s Last Theorem. The Annals of Mathematics. 1995; 141(3): 443. doi: 10.2307/2118559

2. Ossicini A. On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem. Mathematics. 2022; 10(23): 4471. doi: 10.3390/math10234471

3. Klykov SP. Elementary proofs for the Fermat’s last theorem in Z using one trick for a restriction in ZP. Journal of Science and Arts. 2023; 23(3): 603-608. doi: 10.46939/j.sci.arts-23.3-a03

4. Klykov SP, Klykova MV. An elementary proof of Fermat’s last theorem. 2023. doi: 10.13140/RG.2.2.19455.59044

5. Gilbert JB. A Proof of Fermat’s Last Theorem. 2023. doi: 10.13140/RG.2.2.27051.82722

6. Castro C. Finding Rational Points of Circles, Spheres, Hyper-Spheres via Stereographic Projection and Quantum Mechanics. 2023. doi: 10.13140/RG.2.2.12030.36164

7. Carbó-Dorca R. Natural Vector Spaces (inward power and Minkowski norm of a Natural Vector, Natural Boolean Hypercubes) and a Fermat’s Last Theorem conjecture. Journal of Mathematical Chemistry. 2016; 55(4): 914-940. doi: 10.1007/s10910-016-0708-6

8. Carbó-Dorca R, Muñoz-Caro C, Niño A, et al. Refinement of a generalized Fermat’s last theorem conjecture in natural vector spaces. Journal of Mathematical Chemistry. 2017; 55(9): 1869-1877. doi: 10.1007/s10910-017-0766-4

9. Niño A, Reyes S, Carbó-Dorca R. An HPC hybrid parallel approach to the experimental analysis of Fermat’s theorem extension to arbitrary dimensions on heterogeneous computer systems. The Journal of Supercomputing. 2021; 77(10): 11328-11352. doi: 10.1007/s11227-021-03727-2

10. Carbó-Dorca R, Reyes S, Niño A. Extension of Fermat’s last theorem in Minkowski natural spaces. Journal of Mathematical Chemistry. 2021; 59(8): 1851-1863. doi: 10.1007/s10910-021-01267-x

11. Carbó-Dorca R. Whole Perfect Vectors and Fermat’s Last Theorem. Journal of Applied Mathematics and Physics. 2024; 12(01): 34-42. doi: 10.4236/jamp.2024.121004

12. Carbó-Dorca R. Rational Points on Fermat’s Surfaces in Minkowski’s (N+1) -Dimensional Spaces and Extended Fermat’s Last Theorem: Mathematical Framework and Computational Results. Unpublished Preprint. 2023. doi: 10.13140/RG.2.2.34181.52967

13. Carbó-Dorca R. Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling. 2018; 1(1): 1-14. doi: 10.33187/jmsm.413116

14. Carbó-Dorca R. Fuzzy sets and Boolean tagged sets, vector semispaces and convex sets, QSM and ASA density functions, diagonal vector spaces and quantum Chemistry. Adv. Molec. Simil. 1998; 2: 43-72. doi: 10.1016/S1873-9776(98)80008-4

15. Carbó-Dorca R. Role of the structure of Boolean hypercubes when used as vectors in natural (Boolean) vector semispaces. Journal of Mathematical Chemistry. 2019; 57(3): 697-700. doi: 10.1007/s10910-018-00997-9

16. Carbó-Dorca R. Shadows’ hypercube, vector spaces, and non-linear optimization of QSPR procedures. Journal of Mathematical Chemistry. 2021; 60(2): 283-310. doi: 10.1007/s10910-021-01301-y

17. Carbó-Dorca R. Shell partition and metric semispaces: Minkowski norms, root scalar products, distances and cosines of arbitrary order. J. Math. Chem. 2002; 32: 201-223.

18. Bultinck P, Carbó-Dorca R. A mathematical discussion on density and shape functions, vector semispaces and related questions. J. Math. Chem. 2004; 36: 191-200. doi: 10.1023/B:JOMC.0000038793.21806.65

19. Carbó-Dorca R. Molecular quantum similarity measures in Minkowski metric vector semispaces. Journal of Mathematical Chemistry. 2008; 44(3): 628-636. doi: 10.1007/s10910-008-9442-z

20. Carbó-Dorca R, Chakraborty T. Extended Minkowski spaces, zero norms, and Minkowski hypersurfaces. Journal of Mathematical Chemistry. 2021; 59(8): 1875-1879. doi: 10.1007/s10910-021-01266-y

21. Carbó-Dorca R. Generalized scalar products in Minkowski metric spaces. Journal of Mathematical Chemistry. 2021; 59(4): 1029-1045. doi: 10.1007/s10910-021-01229-3

Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Ramon Carbó-Dorca, Debraj Nath

License URL: https://creativecommons.org/licenses/by/4.0/