논문KCI 등재
Fermat's equation over 2-by-2 matrices
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
주제분류수학
초록
We study the solvability of the Fermat's matrix equation in some classes of 2-by-2 matrices. We prove the Fermat's matrix equation has infinitely many solutions in a set of 2-by-2 positive semidefinite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational matrices and stochastic quadratic field matrices.
Abstract
We study the solvability of the Fermat's matrix equation in some classes of 2-by-2 matrices. We prove the Fermat's matrix equation has infinitely many solutions in a set of 2-by-2 positive semidefinite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational matrices and stochastic quadratic field matrices.
참고문헌 (15)
- A. Aigner, "Uber die M¨oglichkeit von x 4 + y 4 = z 4in quadratischen K¨orpern", Jber. Deutsch. Math.-Verein, 43, pp.226~229 (1934)
- Alexander Aigner, "Die Unmöglichkeit von x6+y6=z6 und x9+y9=z9 in quadratischen Körpern", Monatshefte für Mathematik, 61(2), pp.147~150 (1957), Springer Science and Business Media LLC[DOI]
- Zhenfu Cao, "Fermat's type equations in the set of 2×2 integral matrices", Tsukuba Journal of Mathematics, 22(3), pp.637~643 (1998), Institute of Mathematics, University of Tsukuba[DOI]
- R. Z. Domiaty, "Solutions of x 4 + y 4 = z 4 in 2 × 2 Integral Matrices", The American Mathematical Monthly, 73(6), pp.631~ (1966), JSTOR[DOI]
- Nuno Freitas, "Fermat’s last theorem over some small real quadratic fields", Algebra & Number Theory, 9(4), pp.875~895 (2015), Mathematical Sciences Publishers[DOI]
- Aleksander Grytczuk, "On Fermat's equation in the set of integral 2×2 matrices", Periodica Mathematica Hungarica, 30(1), pp.67~72 (1995), Springer Science and Business Media LLC[DOI]
- A. Grytczuk, "Note on the matrix Fermat’s equation", Notes Num. Theory Disc. Math, 17, pp.4~11 (2011)
- R. A. Horn, "Matrix Analysis" (2013), Cambridge University Press
- Frazer Jarvis, "The Fermat equation over Q(√2)", Journal of Number Theory, 109(1), pp.182~196 (2004), Elsevier BV[DOI]
- I. Kaddoura, "A matrix analog to last Fermat’s theorem and Beal’s conjecture", 22th LAAS International Science Conference (2016)
- A. Khazanov, "Fermat’s equation in matrices", Serdica Math. J, 21(1), pp.19~40 (1995)
- Maouha Le, "On Fermat's equation in integral 2×2 matrices", Periodica Mathematica Hungarica, 31(3), pp.219~222 (1995), Springer Science and Business Media LLC[DOI]
- Qin Hourong, "Fermat's problem and Goldbach's problem over MnZ", Linear Algebra and its Applications, 236, pp.131~135 (1996), Elsevier BV[DOI]
- L. N. Vaserstein, "Algebraic K-theory and algebraic number theory", pp.1090~ (1989), Amer. Math. Soc
- Andrew Wiles, "Modular Elliptic Curves and Fermat's Last Theorem", The Annals of Mathematics, 141(3), pp.443~551 (1995), JSTOR[DOI]
관련 논문AI추천
On the Tarry-Escott and Related Problems for 2 × 2 matrices over QSupawadee Prugsapitak, Vichian Laohakosol, Walisa Intarapak경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2023 · 제63권 제3호Solvability of some entangled Diophantine equations박부성대한수학회 · 대한수학회보2014 · 제51권 제5호Linear isomorphisms of non-degenerate integral ternary cubic forms오병권, 이인환대한수학회 · 대한수학회보2016 · 제53권 제6호Conjugacy invariants of quaternion matrices김준형, Qianghua Luo대한수학회 · 대한수학회논문집2020 · 제35권 제4호ON 2 × 2 STRONGLY CLEAN MATRICESHuanyin Chen대한수학회 · 대한수학회보2013 · 제50권 제1호IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS우성식대한수학회 · 대한수학회지2010 · 제47권 제1호
이 간행물의 다른 논문 (17건)
Curvature estimates for gradient expanding Ricci solitonsLiangdi Zhangpp. 537-557Invariant mean value property and $\mathcal M$-harmonicity on the half-space최부림, 남계숙pp. 559-572Polynomiality of the equivariant Gromov-Witten theory of $\mathbb P^{r-1}$노현호pp. 573-591Fourier coefficients of weight 2 modular forms of prime levelYan-Bin Lipp. 593-602Multiplicative functions which are additive on triangular numbers박부성pp. 603-608