# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k + a5k = b1k + b2k + b3k + b4k + b5k
( k = 1, 2, 3, 7 )
• By searching for the positive integer solutions of  c2k + c3k+ c4k+ c5k = d1k + d2k + d3k + d4k + d5( k = 1, 2, 3 ) and by applying Theorem 1, Chen Shuwen obtained the first solution for ( k = 1, 2, 3, 7 ) in 15 Sep 2022.
• [ 261, 816, 821, 1601, 1756 ] = [ 271, 711, 926, 1581, 1766 ]
• Using the same method as above, Chen Shuwen found the following two solutions in September 2022.
• [ 519, 710, 781, 937, 954 ] = [ 521, 694, 807, 909, 970 ]
• [ 41, 148, 248, 389, 502 ] = [ 46, 134, 262, 383, 503 ]
• Smallest solution.
• Summaries on the ( k = 1, 2, ..., n, n+4 ) Series :
• [ 3, 54, 62 ] = [ 24, 28, 67 ]      ( k = 1, 5 )
• Smallest solution, by L.J.Lander, T.R.Parkin and J.L.Selfridge in 1967. ( First known solution was found by A.Moessner in 1939. )
• [ 7, 16, 25, 30 ] = [ 8, 14, 27, 29 ]       ( k = 1, 2, 6 )
• Smallest solution, by Chen Shuwen in 1995-2001.
• [ 41, 148, 248, 389, 502 ] = [ 46, 134, 262, 383, 503 ]       ( k = 1, 2, 3, 7 )
• Smallest solution, by Chen Shuwen in 2022.
• [ -95, -56, 37, 106, 118, 148 ] = [ -92, -62, 49, 85, 136, 142 ]       ( k = 1, 2, 3, 4, 8 )
• Integer solution, not non-negative integer solution, by Chen Shuwen in 2022.
• Chen Shuwen proved that there is no non-negative integer solution in the range of 500.
• Summaries on the ( k = k1, k1+1, k1+2, k1+6 ) Series :
• [ 41, 148, 248, 389, 502 ] = [ 46, 134, 262, 383, 503 ]       ( k = 1, 2, 3, 7 )
• Smallest solution, by Chen Shuwen in 2022.
• [ 205, 248, 255, 344, 351 ] = [ 215, 221, 279, 328, 360 ]     ( k = 0, 1, 2, 6 )
• First known solution, by Chen Shuwen in 2017.

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k + a5k = b1k + b2k + b3k + b4k + b5k
( k = -7, -3, -2, -1 )
• Based on the above first solution of ( k = 1, 2, 3, 7 ) and Theorem 9, Chen Shuwen obtained the first solution of ( k= -7, -3, -2, -1 ) in 15 Sep 2022.
• [ 6293865995771768003448, 7030339878894966662928, 12003204480057173103768, 15632865469103997600688, 41014639662483181897008] = [6329708057251106090028, 6942515520632693500368, 13538328073730745790608, 13621283515358997909423, 42586081795145372774288 ]
• Based on the above smallest solution of ( k = 1, 2, 3, 7 ), Chen Shuwen found the following smallest known solution of ( k = -7, -3, -2, -1 ) in September 2022.
• [ 2840097782598520232, 3729945651819988712, 5452554139874258308, 10660964064530266244, 31055851840153384276 ] = [ 2845755347902501348, 3672414356419166264, 5760359615512321277, 9652494490858484302, 34843150845050138456 ]

Last revised October 3, 2022.