On The Prouhet-Tarry-Escott Problem

The Prouhet-Tarry-Escott problem is a classical problem in number theory.This is the problem of finding two distinct lists of integers[α1,α2…]and[β1,β2,…,βn]such thatα1j+ α2j+ …+αnj=β1j+β2j+…βnj,for j = 1,2,…,k,for some integer 0<k<n.Any pair[α1,α2,…αn],[β1,β2,…,βn]that satisfies the equations is called a solution of the Prouhet-Tarry-Escott problem.We call n the size of the solution and k the degree.The size n must be great,er than or equal to k + 1 has been proved in[18].The solution of degree n-1 and size n is called ideal solution.In 1934,Wright[28]conjectured that there is always an ideal solution for every size.Until now,the ideal solution of size n = 11 and sizes n>13 has not been found.Let[α1,α2,…αn],[β1,β2,…,βn]be an ideal solution of size n,we call Dn = max1≤i<n{αi,βi}-min1≤i<n{αi,βi}the diameter of the solution.We call the ideal solut,ion with smallest diameter the smallest ideal solution of size n.In 2016,Qiu[40]described a method for searching for ideal solutions and idea.l symmetric solutions,and then gave the sma.llest ideal solutions of sizes 3 through 8 and the smallest ideal symmetric solutions of sizes 9 and 12.In this paper,based on Qiu’s algorithm,we describe an improved method for searching for ideal solutions and ideal symmetric solutions of the Prouhet-Tarry-Escott problem by studying their property.And we give a definition of ideal formal symmetric solution and discuss the corresponding algorithm.According to the above algorithms,we give the smallest ideal solution of size 9 and the smallest ideal formal symmetric solution of size 12.On the other hand,we prove that the diameter of the smallest ideal formal symmetric solution of size 16 must be greater than 419.