Ajai Choudhry ; Jaroslaw Wroblewski - Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.

hrj:161 - Hardy-Ramanujan Journal, January 1, 2008, Volume 31 - 2008 - https://doi.org/10.46298/hrj.2008.161
Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.Article

Authors: Ajai Choudhry 1; Jaroslaw Wroblewski 2,3

  • 1 High Commission of India
  • 2 Institut of Mathematics, Wroclaw University
  • 3 University of Wrocław [Poland]

This paper is concerned with the system of simultaneous diophantine equations 6i=1Aki=6i=1Bki for k=2,4,6,8,10. Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree 11, that is, of the system of simultaneous equations, 12i=1aki=12i=1bki for k=1,2,3,,11. We use one of the ideal solutions to prove new results on sums of 13th powers. In particular, we prove that every integer can be expressed as a sum or difference of at most 27 thirteenth powers of positive integers.


Volume: Volume 31 - 2008
Published on: January 1, 2008
Imported on: March 3, 2015
Keywords: Tarry-Escott problem,multigrade equations,thirteenth powers,easier Waring problem,[MATH]Mathematics [math]

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