episciences.org_161_1653161591
1653161591
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
2008
Volume 31  2008
Ideal solutions of the TarryEscott problem of degree eleven with applications to sums of thirteenth powers.
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 wellknown that solutions of the above system lead to ideal solutions of the TarryEscott 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.
01
01
2008
161
https://hal.archivesouvertes.fr/hal01112313v1
10.46298/hrj.2008.161
https://hrj.episciences.org/161

https://hrj.episciences.org/161/pdf