Kyungpook Mathematical Journal

  • 제63권3호
  • /
  • Pages.345-353
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  • 2023
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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On the Tarry-Escott and Related Problems for 2 × 2 matrices over ℚ

  • Supawadee Prugsapitak (Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University) ;
  • Walisa Intarapak (Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University) ;
  • Vichian Laohakosol (Department of Mathematics, Faculty of Science, Kasetsart University)
  • 투고 : 2022.10.31
  • 심사 : 2023.07.05
  • 발행 : 2023.09.30
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초록

Reduced solutions of size 2 and degree n of the Tarry-Escott problem over M2(ℚ) are determined. As an application, the diophantine equation αAn + βBn = αCn + βDn, where α, β are rational numbers satisfying α + β ≠ 0 and n ∈ {1, 2}, is completely solved for A, B, C, D ∈ M2(ℚ).

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과제정보

We would like to express our gratitude to the referee(s) for comments and suggestions.

참고문헌

  1. A. Alpers and R. Tijdeman, The two-dimensional Prouhet-Tarry-Escott problem, J. Number Theory, 123(2)(2007), 403-412. https://doi.org/10.1016/j.jnt.2006.07.001
  2. P. Borwein and C. Ingalls, The Prouhet-Tarry-Escott problem revisited, Enseign. Math., 40(2)(1994), 3-27.
  3. A. Choudhry, Matrix analogues of the Tarry-Escott problem, multigrade chains and the equation of Fermat, Math. Student, 75(2006), 215-224.
  4. B. Cohen, Generalized Pell Equation for 2 × 2 matrices, Discuss. Math. GAA, 37(2017), 13-30. https://doi.org/10.7151/dmgt.1916
  5. H. L. Dorwart and O. E. Brown, The Tarry-Escott Problem, Amer. Math. Monthly, 44(10)(1937), 613-626. https://doi.org/10.1080/00029890.1937.11988044
  6. L. N. Vaserstein, Noncommutative number theory, in Algebraic K-theory and Algebraic Number Theory, Contemp. Math. 83, Amer. Math. Soc., (1989), 445-449. https://doi.org/10.1090/conm/083/991989