Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.

Abstract

International audience 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.

Equal Sums of Like Powers with Minimum Number of Terms

“…52, 55-58], [9], [11, pp. 41-54], [12], [14]) as well as for k = 11 [5]. Thus, for these values of k, we have β(k) ≤ 2k + 1, and in particular, we get…”

The Prouhet-Tarry-Escott problem for Gaussian integers

“…We are currently working on another computational method which would generalize the work of Smyth in [23] and Choudhry and Wróblewski in [11]. Both of these articles relate the Z-pte problem to elliptic curves.…”

“…They were both found computationally, by Kuosa, Myrignac and Shuwen [22] and Broadhurst [6]. However, in 2008, Choudhry and Wróblewski [11] found some infinite inequivalent families of solutions for n = 12 (again incomplete). Both infinite families of solutions for sizes 10 and 12 arise from rational points on elliptic curves using a method of Letac's from 1934, which appears in [15].…”

“…Table 2, which largely comes from [4,5] but is updated with the new solutions from [6] and [11], shows that the constant for the Z-pte problem has many more factors than that for the Z[i]pte problem. This demonstrates that the Gaussian (…”

New solutions of the Tarry-Escott problem of degrees 2, 3 and 5

“…It would be recalled that simple solutions of the TEP of degree 2 were first noticed by Goldbach and by Euler in 1750-51. Ever since then, numerous authors have given parametric ideal solutions of the TEP when 2 ≤ k ≤ 7 and numerical ideal solutions when k = 8, 9 or 11 ( [1], [3], [4], [5], [6, pp. 705-713], [8, pp.…”