On Fermat's Last Theorem Matrix Version and Galaxies of Sequences of Circulant Matrices with Positive Integers as Entries
Fermat's equation, polynomials, model theory, circulant matrices, Mouanda's choise function, galaxy, Toeplitz matrices
Abstract
We introduce Mouanda's choice function for matrices which allows us to construct the galaxies of sequences of triples of circulant matrices with positive integers as entries. We give many examples of the galaxies of circulant matrices with positive integers as entries. The characterization of the matrix solutions of the equation allows us to show that the equation has no circulant matrix with positive integers as entries solutions. This allows us to prove that, in general, the equation has no circulant matrix with positive integers as entries solutions. We prove Fermat's Last Theorem for eigenvalues of circulant matrices. Also, we show Fermat's Last Theorem for complex polynomials over associated to circulant matrices.
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Alexandria Diophantus (2014) Diophantus of Alexandria and Arithmetica. 62-69.
L Euler (2020) E-98: Theorematum quorundam arithmeticorum demonstrationes. 10, 284-292.
L Euler (1770) Vollstandige Anleitung zur Algebra.
A Legendre (1823) Recherches sur quelques objets d'analyse indeterminee, et particulierement sur the theoreme de Fermat. 1-60.
A Legendre (1830) Didot, Firmin. II.
P Dirichlet (1832) Démonstration du théorème de Fermat pour le cas des 14ièmes puissances.. 1832(9), 390-393.
G Lame (1839) THÉORIE DES NOMBRES. - Mémoire sur de nouvelles formules relatives à la théorie des polynômes radicaux, et sur le dernier theoreme de Fermat.. 9, 240-254.
G Lame (1840) Memoire d'analyse indeterminee demontrant que l'equation x 7 + y 7 = z 7 est impossible en nombres entiers. 5, 195-211.
G Lame (1847) Memoire sur la resolution en nombres complexe de l'equation A 5 + B 5 + C 5 = 0. 12, 137-171.
G Lame (1865) Comptes Rendus Hebdomadaires des Seances de L'Academie des Sciences. 8(30), 27-27.
V Lebesque (1840) Demonstration de l'impossibilite de resoudre l'equation x 7 + y 7 + z 7 = 0 est impossible en nombres entiers. 5, 348-349.
V Lebesque (1843) Theoremes nouveaux sur l'equation indeterminee x 5 +y 5 = az 5. 70(3-4), 49-70.
G Frey (1986) Links between stable elliptic curves and certain Diophantine equations. 1, 1-40.
Jean-Pierre Serre (1972) Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques. 15(4), 259-331.
Jean-Pierre Serre (1987) Sur les représentations modulaires de degré 2 de Gal(Q¯/Q). 54(1), 170-230.
J Serre (1996) Travaux de Wiles (et Taylor). 237, 319-332.
K Ribet (1990) On modular representations of $$(bar Q/Q)$$ arising from modular forms. 100(1), 431-476.
B Mazur (1987) Deforming Galois Representations. 16, 385-437.
Barry Mazur (1977) Modular curves and the Eisenstein ideal. 47, 33-186.
Richard Taylor, Andrew Wiles (1995) Ring-Theoretic Properties of Certain Hecke Algebras. 141(3), 553.
Andrew Wiles (1995) Modular Elliptic Curves and Fermat's Last Theorem. 141(3), 443.
C Breuil, B Conrad, F Diamond, R Taylor (2001) On the modularity of elliptic curves over : wild 3-adic exercices. 14(4), 843-939.
Brian Conrad, Fred Diamond, Richard Taylor (1999) Modularity of certain potentially Barsotti-Tate Galois representations. 12(2), 521-567.
Fred Diamond (1996) On Deformation Rings and Hecke Rings. 144(1), 137.
Bibek Nag (2021) An Elementary Proof of Fermat's Last Theorem for Epsilons. 11(08), 735-740.
Joachim Mouanda (2022) On Fermat's Last Theorem and Galaxies of Sequences of Positive Integers. 12(01), 162-189.
R Domiaty (1966) Solutions of x 4 + y 4 = z 4 in 2 × 2 Integral Matrices. 73(6), 631.
L Veserstein (1989) Noncommutative number theory. 83, 445-449.
D Frejman (1993) On Fermat's in the set of Fibonacci matrices. 13, 61-64.
Aleksander Grytczuk (1995) On Fermat's equation in the set of integral 2×2 matrices. 30(1), 67-72.
A Khazanov (1995) Fermat's equation in matrices. 21, 19-40.
H Qin (1996) Fermat's problem and Goldbach problem over Mn( ). 236, 131-135.
Z Patay, A Szakacs (2002) On Fermat's problem in matrix ring and groups. 61(3-4), 487-494.
Mao-Ting Chien, Jie Meng (2021) Fermat's equation over 2-by-2 matrices. 58(3), 609-616.
Enrico Laeng (1999) On Fermat's Last Theorem for Polynomials. 35(1).
P Davis (1979) John Wiley & Sons, Inc.. 51(2), 265A-265A.
P Davis (1994) Circulant Matrices.
P Davis (2013) Circulant Matrices. (2).
D Geller, I Kra, S Popescu, S Simanca (2002) On circulant matrices.
Walter Rudin (1962) Fourier Analysis on Groups.
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2022-06-01
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