## Results and discussion on the k < 0 case

a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = k1 , k2 , ... , kn with k1 <0)

• Introduction
• 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 ]
• In Guo Xianqiang's site on Equal sums of like powers ( in Chinese), he ask such an interesting question :
• If k1< 0 and kn > 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 , ... , kn ), solutions can be easily obtained by simply transforming from a corresponding known type:
• [ a1 , a2 , ... , am ] = [ b1 ,b2 , ... , bm       ( k = k1, k2 , ... , kn )
<=>    [ C/a1 , C/a2 , ... , C/am ] = [ C/b1 ,C/b2 , ... , C/bm ]      ( k = -k1, -k2 , ... , -kn )
where C is the least common multiple of all { ai , bi }
• 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 kn > 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 kn > 0:
• a1k + a2k + ... + amk = b1k + b2k + ... + bmk     ( k = k1 , k2 , ... , kn )

• Positive integer solutions
• Here we consider the k1< 0 , kn > 0 and m=n+1 case:
a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k     ( k = k1 , k2 , ... , kn )
This system will be denoted as:
[ a1 , a2 , ... , an+1 ] = [ b1 ,b2 , ... , bn+1       ( k = k1 , k2 , ... , kn )
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.

• Discussion
• Chen Shuwen have checked at least 250 sets of solutions of ( k = -1, 1 ). All of them satisfy the following relation:
• 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

Last revised May 6, 2001.
Copyright 1997-2001, Chen Shuwen