*a*_{1}^{k}
+ a_{2}^{k} + ... + a_{m}^{k}
= b_{1}^{k} + b_{2}* ^{k}
+ ... + b_{m}^{k}*
(

- 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
*k*_{1}< 0 and*k*> 0 , is there solution in positive integers?_{n } - Guo Xianqiang guessed the answer is NO. However, Chen Shuwen solved ( k = -1, 1 ) in May of 2001.
- In April of 2001, Chen Shuwen pointed out that,
if all the
*k<=*0, (*k*=*k*_{1},*k*_{2}*k*), solutions can be easily obtained by simply transforming from a corresponding known type:_{n} - [
*a*_{1}*a*_{2}*a*]_{m}*b*_{1}*,b*_{2}, ... ,*b*]_{m}*k*=*k*_{1},*k*_{2}*k*)_{n} - <=> [
*C/a*_{1}*C/a*_{2}*C/a*]_{m}*C/b*_{1}*,C/b*_{2}, ... ,*C/b*] (_{m}*k*= -*k*_{1},*-k*_{2}*k*)_{n} - where
*C*is the least common multiple of all {*a*,_{i}*b*}_{i} - For example, from the following solution of (
*k*= 1, 2, 4, 6 ): - [ 3, 7, 10, 16, 16 ] = [ 4, 5, 12, 14, 17 ]
- We can get a solution of (
*k*= -1, -2, -4, -6 ) without any difficult ( here C=28560 ): - [ 1680, 2040, 2380, 5712, 7140 ] = [ 1785, 1785, 2856, 4080, 9520 ]
- The above transformation is also true for the
case of
*k*_{1}< 0 and*k*> 0. For example::_{n } - [
*a*_{1 },*a*_{2}*a*_{3}*b*_{1 },*b*_{2 },*b*_{3}*k*= -1, 2 ) - <=> [
*C/a*_{1}*C/a*_{2}*C/a*_{3}]*C/b*_{1}*,C/b*_{2},*C/b*_{3}] (*k*= -2, 1 ) - In this page, we will only consider the following
system in the case of
*k*_{1}< 0 and*k*> 0:_{n } *a*_{1}^{k}+ a_{2}^{k}+ ... + a_{m}^{k}= b_{1}^{k}+ b_{2}(^{k}+ ... + b_{m}^{k}*k*=*k*_{1}*k*_{2}*k*)_{n}

- Here we consider the
*k*_{1}< 0 ,*k*> 0 and_{n }*m*=*n*+1 case: *a*_{1}^{k}+ a_{2}^{k}+ ... + a_{n}_{+1}^{k}= b_{1}^{k}+ b_{2}^{k}+ ... + b_{n}_{+1}(^{k}*k*=*k*_{1}*k*_{2}*k*)_{n}- This system will be denoted as:
- [
*a*_{1}*a*_{2}*a*_{n}_{+1}*b*_{1}*,b*_{2}, ... ,*b*_{n}_{+1}*k*=*k*_{1}*k*_{2}*k*)_{n} - Positive integer solutions have been found to the following 1 type of this system:
**( k = -1, 1 )**- [ 4, 10, 12 ] = [ 5, 6, 15 ]
- Smallest solution, by Chen Shuwen in 2001.
- [ 6, 14, 14 ] = [ 7, 9, 18 ]
- [ 3, 40 ] = [ 4, 15, 24 ] = [ 5, 8, 30 ]
- By Chen Shuwen in 2001.

- Chen Shuwen have checked at least 250 sets of solutions of ( k = -1, 1 ). All of them satisfy the following relation:
- a1 < b1 <= b2 < a2 <= a3 < b3
- See also Conjectures by Chen Shuwen.
- It should be noticed that [
*a*_{1},*a*_{2}]*b*_{1},*b*_{2},*b*_{3}] of ( k = -1, 1 ) can not be denoted as [ 0,*a*_{1},*a*_{2}]*b*_{1},*b*_{2},*b*_{3}]. - It is very easy to prove that the following system has no non-trivial solution in integer.
*a*_{1}*a*_{2}*b*_{1}*+ b*_{2}- 1
`/`*a*_{1}`/`*a*_{2}`/`*b*_{1}*+*1`/`*b*_{2}

Copyright 1997-2001, Chen Shuwen