# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k + a5k + a6k + a7k = b1k + b2k + b3k + b4k + b5k + b6k + b7k
( k = 1, 2, 3, 5, 7, 9 )
• Chen Shuwen obtained the first known solution for ( k = 1, 2, 3, 5, 7, 9 ) in 27 Jan 2023.
• [ 7, 89, 91, 251, 253, 341, 373 ] = [ 29, 31, 151, 193, 311, 313, 377 ]
• By computer searching, Chen Shuwen proved that it is the smallest solution.
• [ 269, 397, 409, 683, 743, 901, 923 ] = [ 299, 313, 493, 613, 827, 839, 941 ]
• Second known solution, by Chen Shuwen in 6 Mar 2023.
• Summaries on the ( k = 1, 2, 3, 5, ..., 2n-1, 2n+1 ) Series :
• [ 15, 25, 55, 55, 73] = [ 13, 31, 43, 67, 69 ]
•       ( k = 1, 2, 3, 5 )
• First known solution, by Chen Shuwen in 1995.
• [ 87, 233, 264, 396, 496, 540 ] = [ 90, 206, 309, 366, 522, 523 ]
•      ( k = 1, 2, 3, 5, 7 )
• First known solution, by Chen Shuwen in 1999.
• [ 7, 89, 91, 251, 253, 341, 373 ] = [ 29, 31, 151, 193, 311, 313, 377 ]      ( k = 1, 2, 3, 5, 7, 9 )
• First known solution, smallest solution, by Chen Shuwen in 2023.

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k + a5k + a6k + a7k = b1k + b2k + b3k + b4k + b5k + b6k + b7k
( k = -9, -7, -5, -3, -2, -1 )
• Based on the above first solution of ( k = 1, 2, 3, 5, 7, 9 ) and Theorem 9, Chen Shuwen obtained the first solution of ( k = -9, -7, -5, -3, -2, -1 ) in 27 Jan 2023.
• [ 1297752271862287613603, 1563107368984288914787, 1573159506405409743821, 2534987598404572177867, 3240083486702532651181, 15782342144905884849301, 16870779534209738976839 ] = [ 1311669186305851019647, 1434758376809625895391, 1933804768743408815527, 1949213571681603308081, 5376402269143762970641, 5497220297663847531779, 69893229498868918618333 ]

Last revised April 16, 2023.

Copyright 1997-2023, Chen Shuwen