On galaxies of sequences of toeplitz matrix solutions of the Diophantine Equation X n  + Y n  = Z n , n ≥ 3

If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary diﬀerential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we ﬁnd that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.

Fermat surfaces and hypercubes

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.

On galaxies of sequences of toeplitz matrix solutions of the Diophantine Equation Xⁿ + Yⁿ = Zⁿ, n ≥ 3

Joachim Moussounda Mouanda

Article ID: 2623
Vol 2, Issue 1, 2024

DOI: https://doi.org/10.54517/mss.v2i1.2623

Received: 23 January 2024; Accepted: 19 April 2024; Available online: 28 April 2024; Issue release: 30 June 2024

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Abstract

We construct the galaxies of sequences of Toeplitz matrix solutions of the Diophantine equation Xⁿ + Yⁿ = Zⁿ, n ≥ 3, linked to Pythagorean triples.

Keywords

matrices of integers; solutions of Diophantine equations; Fermat’s equations

References

Moussounda Mouanda J. On Fermat’s Last Theorem and Galaxies of sequences of positive integers, American Journal of Computational Mathematics. 2022; 12: 162-189. Diophantus of Alexandria, Arithmetica of Diophantus, 1637. Wiles A. Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics. 1995; 141(3): 443-551. Nag BB. An elementary proof of Fermat’s Last Theorem for epsilons. Advances in Pure Mathematics. 2021; 11: 735-740. Domiaty RZ. Solution of X4 + Y4 = Z4 in integral matrices. Amer. Math. Monthly. 1966; 73: 631. Veserstein LN. Non-commutative Number Theory. Contemp. Math. 1989; 83: 445-449. Frejman D. On Fermat’s in the set of Fibonacci matrices. Discuss. Math. 1993; 13: 61-64. Grytczuk A. On Fermat’s equation in the set integral 2 × 2 matrices. Period. Math. Hung. 1995; 30: 79-84. Khazanov A. Fermat’s equation in matrices. Serdica Math. J. 1995; 21: 19-40. Qin H. Fermat’s problem and Goldbach problem over Mn(Z). Linear Algebra Appl. 1996; 236: 131-135. Patay Z and Szakacs A. On Fermat’s problem in matrix ring and groups. Publ. Math. Debrecen. 2022; 61(3-4): 487-494. Chien M-T and Meng J. Fermat’s equation over 2-by-2 matrices. Bull. Korean Math. Soc. 2021; 58(3): 609-616. Laeng E. On Fermat’s Last Theorem for Polynomials. Parabola 1999; 35(1). Moussounda Mouanda J. On Fermat’s Last Theorem Matrix Version and Galaxies of Sequences of Circulant Matrices with Positive Integers as Entries. Global Journal of Science Frontier Research. 2022; 22(2): 37-65. Moussounda Mouanda J. On Beal’s Conjecture for Matrix Solutions and Multiplication Commutative Groups of Rare Matrices. Turkish Journal of Analysis and Number Theory. 2024; 12(1), 1-7.

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