# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k + a5k + a6k = b1k + b2k + b3k + b4k + b5k + b6k
( k = 1, 2, 3, 4, 7 )
• By searching for the positive integer solutions of  c2k + c3k + c4k + c5k + c6k= d1k + d2k + d3k + d4k + d5+ d6 ( 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

a1k + a2k + a3k + a4k + a5k + a6k = b1k + b2k + b3k + b4k + b5k + b6k
( 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.