{"docId":161,"paperId":161,"url":"https:\/\/hrj.episciences.org\/161","doi":"10.46298\/hrj.2008.161","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":42,"name":"Volume 31 - 2008"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01112313","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-01112313v1","dateSubmitted":"2015-03-03 16:14:00","dateAccepted":"2015-06-12 16:05:55","datePublished":"2008-01-01 08:00:00","titles":{"en":"Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers."},"authors":["Choudhry, Ajai","Wroblewski, Jaroslaw"],"abstracts":{"en":"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."},"keywords":[["Tarry-Escott problem"],["multigrade equations"],["easier Waring problem"],["thirteenth powers"],"[MATH] Mathematics [math]"]}