# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k+ a5k+ a6k+ a7k + a8k = b1k + b2k + b3k + b4k + b5k + b6k + b7k+ b8k
( k = 1, 2, 3, 4, 5, 6, 8 )
• Chen Shuwen found one non-negative integer solution in 1999, with the help of a Pentium 100 PC.
• [ 77, 159, 169, 283, 321, 443, 447, 501 ] = [ 79, 137, 213, 237, 363, 399, 481, 491 ]
• T.N.Sinha conjectured that the system
• a1k + a2k + ... + ank = b1k + b2k + ... + bnk      ( k = 1, 2, ..., j-1 , j+1, ..., n
has nontrivial solution in positive integer for all n. [6] [32]
To the special case j = n -1, that is
a1k + a2k + ... + ank = b1k + b2k + ... + bnk      ( k = 1, 2, ..., n-2, n
We can prove that Sinha's conjecture is true for n<=8 ( j = n -1 ) by the following known results
• [ 2, 10, 12 ] = [ 3, 8, 13 ]     ( k = 1, 3 )
• [ 2, 7, 11, 15 ] = [ 3, 5, 13, 14 ]     ( k = 1, 2, 4 )
• [ 1, 8, 13, 24, 27 ] = [ 3, 4, 17, 21, 28 ]     ( k = 1, 2, 3, 5 )
• [ 11, 23, 24, 47, 64, 70 ] = [ 14, 15, 31, 44, 67, 68 ]     ( k = 1, 2, 3, 4, 6 )
• [ 43, 169, 295, 607, 667, 1105, 1189 ] = [ 79, 97, 379, 505, 727, 1093, 1195 ]    ( k = 1, 2, 3, 4, 5, 7 )
• [ 77, 159, 169, 283, 321, 443, 447, 501 ] = [ 79, 137, 213, 237, 363, 399, 481, 491 ]

Last revised March,31, 2001.