Solutions to the Multi-dimensional Equal Powers Problem Constructed by Composition of Rectangular Morphisms

  • Anton Černý
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)


Based on the original approach of Eugène Prouhet, using composition of morphism-like array-words mappings, we provide a construction of solutions to the multi-dimensional Prouhet-Tarry-Escott problem.


Prouhet-Tarry-Escott problem array word spectrum symbol position 


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  1. 1.
    Allouche, J.P., Shallit, J.O.: Automatic sequences - theory, applications, generalizations. Cambridge University Press (2003)Google Scholar
  2. 2.
    Alpers, A., Tijdeman, R.: The two-dimensional Prouhet-Tarry-Escott problem. J. Number Theory 123(2), 403–412 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berstel, J., Perrin, D.: The origins of combinatorics on words. Eur. J. Comb. 28(3), 996–1022 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Černý, A., Gruska, J.: Modular trellises. In: The Book of L, pp. 45–61. Springer, Heidelberg (1986)Google Scholar
  5. 5.
    Černý, A.: Generalizations of Parikh mappings. RAIRO Theor. Inform. Appl. 44(2), 209–228 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Černý, A.: On Prouhet’s solution to the equal powers problem. Theoretical Computer Science 491(17), 33–46 (2013)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Černý, A., Gruska, J.: Modular real-time trellis automata. Fundamenta Informaticae IX, 253–282 (1986)Google Scholar
  8. 8.
    Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 329–438. Springer (1997)Google Scholar
  9. 9.
    Dudík, M., Schulman, L.J.: Reconstruction from subsequences. J. Comb. Theory Ser. A 103, 337–348 (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Frougny, C., Vuillon, L.: Coding of two-dimensional constraints of finite type by substitutions. Journal of Automata, Languages and Combinatorics 10(4), 465–482 (2005)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kalashnik, L.: The reconstruction of a word from fragments. In: Numerical Mathematics and Computer Technology. Akad. Nauk Ukrain. SSR Inst. Mat., pp. 56–57 (1973) (in Russian)Google Scholar
  12. 12.
    Knuth, D.E.: The art of computer programming, 2nd edn. Seminumerical algorithms, vol. 2. Addison-Wesley (1981)Google Scholar
  13. 13.
    Krasikov, I., Roditty, Y.: On a reconstruction problem for sequences. J. Comb. Theory, Ser. A 77(2), 344–348 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lehmer, D.H.: The Tarry-Escott problem. Scripta Mathematica 13, 37–41 (1947)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Lothaire, M.: Combinatorics on words. Cambridge University Press (1997)Google Scholar
  16. 16.
    Milner, R.: The spectra of words. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity. LNCS, vol. 3838, pp. 1–5. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. Journal d’Analyse Mathamatique 53, 139–186 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Prouhet, E.: Mėmoire sur quelques relations entre les puissances des nombres. C.R. Acad. Sci. Paris 33, 255 (1851)Google Scholar
  19. 19.
    Rigo, M., Maes, A.: More on generalized automatic sequences. Journal of Automata, Languages and Combinatorics 7(3), 351–376 (2002)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Anton Černý
    • 1
  1. 1.Department of Information Science, College of Computing Sciences and EngineeringKuwait UniversityKuwait

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