`Equal Sums of Like Powers`

Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k+ a₄^k+ a₅^k+ a₆^k+ a₇^k + a₈^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k+ b₈^k

( 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 ]

a₁^k + a₂^k + ... + a_n^k = b₁^k + b₂^k + ... + b_n^k ( 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

a₁^k + a₂^k + ... + a_n^k = b₁^k + b₂^k + ... + b_n^k ( 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 ]