2021/06/26 by Ajai Choudhry, Choudhry, Ajai
Computer Science · Mathematics · #11D09 #11D25 #11D41 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2106.13944
openalex publication_date 2021/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we obtain new parametric ideal solutions of the Tarry-Escott problem of degrees 2, 3 and 5, that is, of the diophantine systems ∑i=1k+1xij=∑i=1k+1yij, j=1, 2, …, k, when k is 2, 3 or 5. When k=2, we obtain the complete ideal solution in terms of polynomials in six parameters p, q, r, a, b and c such that the common sums σj=∑i=13xij=∑i=13yij for both j=1 and j=2 are symmetric functions of the parameters p, q, r and also symmetric functions of the parameters a, b, c. When k=3, we obtain a solution in terms of polynomials in four parameters p, q, r and s such that the three common sums σj= ∑i=14xij=∑i=14yij, j=1, 2, 3, are symmetric functions of all the four parameters p, q, r and s. When k=5, our solution is derived from the solution already obtained when k=2, and the common sums, defined as in the cases when k=2 or 3, are either 0 or have properties similar to the case when k=2.