`Equal Sums of Like Powers`

Non-negative Integer Solutions of

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

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

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

[ 266, 494, 494, 1463, 1547 ] = [ 287, 374, 611, 1394, 1598 ]

Summaries on the ( k = -1, 1, 2, ..., n ) Series :
- [ 4, 10, 12 ] = [ 5, 6, 15 ] ( k = -1, 1 )
  - First known solution, smallest solution, by Chen Shuwen in 2001.
- [ 15, 28, 48, 70 ] = [ 16, 24, 55, 66 ] ( k = -1, 1, 2 )
  - First known solution, smallest solution, by Chen Shuwen in 2017.
- [ 266, 494, 494, 1463, 1547 ] = [ 287, 374, 611, 1394, 1598 ] ( k = -1, 1, 2, 3 )
  - First known solution, by Chen Shuwen in 2022.

Non-negative Integer Solutions of

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

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

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

[ 779779, 893893, 2039422, 3331783, 4341766 ] = [ 805486, 851734, 2522443, 2522443, 4684537 ]

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