郭先强和陈漱文的合作研究

• 导言
  • The Prouhet-Tarry-Escott problem can be stated as:

    Given a positive integer n, find two sets of integer solutions { a₁, a₂, ... , a_m } and { b₁, b₂, ... , b_m } such that the integers in each set have the same sum, the same sum of squares, etc., up to and including the same sum of n^th powers, i.e., we are to find solutions in integers of the system of equations

    a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = 1, 2, ..., n ) 

  • Solutions of this system will be denoted here by the notation

    [ a₁ , a₂ , ... , a_m ] = [ b₁ , b₂ , ... , b_m ]       ( k = 1, 2, ..., n ) 

  • A good online reference by Peter Borwein and Colin Ingalls: The Prouhet-Tarry-Escott Problem Revisited. [44]

• 介绍
  • Guo Xianqiang studied the k < 0 cases of the Equal sums of like powers system first. He gave some numerial examples such as the following solution for ( k = 0, -1, -2, -3, -4 ) before march of 2001:

    [684, 855, 1140, 2160, 4104, 4560, 20520 ] = [ 720, 760, 1368, 2052, 2565, 10260, 13680 ]  

    • Note: Here k = 0 is used to denote the equal products equation. Please refer to Equal products and equal sums of like powers for detail.

  • In Guo Xianqiang's site on Equal sums of like powers ( in Chinese), he ask such an interesting question :

    If k₁< 0 and k_n > 0 , is there solution in positive integers?

    Guo Xianqiang guessed the answer is NO. However, Chen Shuwen solved ( k = -1, 1 ) in May of 2001.

  • In April of 2001, Chen Shuwen noticed that, if all the k<=0,  ( k = k₁, k₂ , ... , k_n ), solutions can be easily obtained by simply transforming from a corresponding known type:

    [ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ]       ( k = k₁, k₂ , ... , k_n )

    <=>    [ C/a₁ , C/a₂ , ... , C/a_m ] = [ C/b₁ ,C/b₂ , ... , C/b_m ]      ( k = -k₁, -k₂ , ... , -k

• 资料介绍

  The Equal Products and Equal Sums of Like Powers is the system of the form

  a₁a₂ ... a_m = b₁ b₂ ... b_m

  a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = k₂ , k₃ , ... , k_n ) 

  This system has been studied for a long time when k <= 3.[13] [5] [17] [6] [59]

  In March of 2001, for the first time, Guo Xianqiang and Chen Shuwen found that the equal products equation can be regarded as the k = 0 case of the Equal Sums of Like Powers system.

  In these pages, we will denote the Equal Products and Equal Sums of Like Powers system as:

  [ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ]       ( k = 0, k₂ , k₃ , ... , k_n</