Equal Sums of Like Powers

Results and discussion on the k < 0 case

• Introduction
• Positive integer solutions
• Discussion

• 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 pointed out 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_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 k₁< 0 and k_n > 0. For example::
[ a₁ , a₂ , a₃ ] = [ b₁ , b₂ , b₃ ]       ( k = -1, 2 )
<=>    [ C/a₁ , C/a₂ , C/a₃ ] = [ C/b₁ ,C/b₂ , C/b₃ ]      ( k = -2, 1 )
• In this page, we will only consider the following system in the case of k₁< 0 and k_n > 0:
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k     ( k = k₁ , k₂ , ... , k_n )

Here we consider the k₁< 0 , k_n > 0 and m=n+1 case:
a₁^k + a₂^k + ... + a_n+1^k = b₁^k + b₂^k + ... + b_n+1^k     ( k = k₁ , k₂ , ... , k_n )
This system will be denoted as:
[ a₁ , a₂ , ... , a_n+1 ] = [ b₁ ,b₂ , ... , b_n+1 ]       ( k = k₁ , k₂ , ... , 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 [ a₁, a₂ ] = [ b₁, b₂ , b₃ ] of ( k = -1, 1 ) can not be denoted as [ 0, a₁, a₂ ] = [ b₁, b₂ , b₃ ].
• It is very easy to prove that the following system has no non-trivial solution in integer.
a₁ + a₂ = b₁ + b₂
1/a₁ + 1/a₂ = 1/b₁ + 1/b₂