10.46298/hrj.2008.161
https://hrj.episciences.org/161
Choudhry, Ajai
Ajai
Choudhry
Wroblewski, Jaroslaw
Jaroslaw
Wroblewski
Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.
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.
episciences.org
Tarry-Escott problem
multigrade equations
easier Waring problem
thirteenth powers
[MATH] Mathematics [math]
2015-06-12
2008-01-01
2008-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01112313v1
2804-7370
https://hrj.episciences.org/161/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 31 - 2008
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