`Equal Sums of Like Powers`

Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k + a₄^k + a₅^k + a₆^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k

( k = 1, 2, 3, 4, 7 )

By searching for the positive integer solutions of c₂^k + c₃^k + c₄^k+ c₅^k + c₆^k= d₁^k + d₂^k + d₃^k + d₄^k + d₅^k+ d₆^k ( k = 1, 2, 3, 4 ) and by applying Theorem 1, Chen Shuwen obtained the first solution for ( k = 1, 2, 3, 4, 7 ) in 22 Sep 2022.

[ 34, 133, 165, 299, 332, 366 ] = [ 35, 124, 177, 286, 353, 354 ]
- By computer searching, Chen Shuwen proved that it is the smallest solution.

Summaries on the ( k = 1, 2, ..., n, n+3 ) Series :
- [ 3, 25, 38 ] = [ 7, 20, 39 ] ( k = 1, 4 )
  - First known solution, smallest solution, by Chen Shuwen in 1995.
- [ 1, 28, 39, 58 ] = [ 8, 14, 51, 53 ] ( k = 1, 2, 5 )
  - Smallest solution, by Chen Shuwen in 1995-2001.
- [ 7, 18, 55, 69, 81 ] = [ 9, 15, 61, 63, 82 ] ( k = 1, 2, 3, 6 )
  - First known solution, smallest solution, by Chen Shuwen in 1999.
- [ 34, 133, 165, 299, 332, 366 ] = [ 35, 124, 177, 286, 353, 354 ] ( k = 1, 2, 3, 4, 7 )
  - First known solution, smallest solution, by Chen Shuwen in 2022.

Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k + a₄^k + a₅^k + a₆^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k

( k = -7, -4, -3, -2, -1 )

Based on the above first solution of ( k = 1, 2, 3, 4, 7 ) and Theorem 9, Chen Shuwen obtained the first solution of ( k = -7, -4, -3, -2, -1 ) in 22 Sep 2022.

[ 3985023979526590, 4393128844899795, 4877989219086060, 8839507736404436, 10966306590276180, 42897611073727410 ] = [ 4120109538154610, 4131781236562980, 5099716001771790, 8240219076309220, 11762248197634935, 41671965043049484 ]

Last revised September 23, 2022.
Copyright 1997-2022, Chen Shuwen