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 - 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.

Authors: Ajai Choudhry ; Jaroslaw Wroblewski

This paper is concerned with the system of simultaneous diophantine equations $\sum_{i=1}^6A_i^k=\sum_{i=1}^6B_i^k$ 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, $\sum_{i=1}^{12}a_i^k=\sum_{i=1}^{12}b_i^k$ for $k=1, 2, 3,\ldots,11.$ We use one of the ideal solutions to prove new results on sums of $13^{th}$ 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
Published on: January 1, 2008
Submitted on: March 3, 2015
Keywords: Tarry-Escott problem, multigrade equations, easier Waring problem, thirteenth powers,[MATH] Mathematics [math]


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