From: lisonek@cecm.sfu.ca (Petr Lisonek)
Subject: New size 10 solutions of the Prouhet-Tarry-Escott Problem
Date: 22 Jun 00 03:44:24 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

The following is the smallest ideal symmetric solution
of the Prouhet-Tarry-Escott problem of type (k = 1, 2, 3, 4, 5, 6, 7, 8, 9):

A1:=[-313, -301, -188, -100, -99, 99, 100, 188, 301, 313];
B1:=[-308, -307, -180, -131, -71, 71, 131, 180, 307, 308];

A second solution is given by

A2:=[-515, -452, -366, -189, -103, 103, 189, 366, 452, 515];
B2:=[-508, -471, -331, -245, -18, 18, 245, 331, 471, 508];

This means that the 10-tuples (Ai,Bi) satisfy

  a1^k + a2^k + a3^k + a4^k + a5^k + a6^k + a7^k + a8^k + a9^k + a10^k
= b1^k + b2^k + b3^k + b4^k + b5^k + b6^k + b7^k + b8^k + b9^k + b10^k

for  k = 1, 2, 3, 4, 5, 6, 7, 8, 9.


The previous smallest known solution found by A. Letac in the 1940s is

A3:=[-23750, -20667, -20449, -11857, -436, 436, 11857, 20449, 20667, 23750];
B3:=[-23738, -20885, -20231, -11881, -12, 12, 11881, 20231, 20885, 23738];

For further details on this problem see

P. Borwein & C. Ingalls, "The Prouhet-Tarry-Escott Problem Revisited"
Ens. Math. 40 (1994) 3-27  (http://www.cecm.sfu.ca/~pborwein/PAPERS/P98.pdf)

or http://member.netease.com/~chin/eslp/TarryPrb.htm

This is the result of a significant amount of non-trivial and ongoing
searching and will be discussed in a future paper.


Peter Borwein, Petr Lisonek and Colin Percival
Simon Fraser University