`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 + a₉^k+ a₁₀^k+ a₁₁^k+ a₁₂^k

= b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k+ b₉^k+ b₁₀^k+ b₁₁^k+ b₁₂^k

( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )

Nuutti Kuosa discovered the following solution in 3 Sep1999, using a program written by Jean-Charles Meyrignac.

151¹⁰+140¹⁰+127¹⁰+86¹⁰+61¹⁰+22¹⁰=148¹⁰+146¹⁰+121¹⁰+94¹⁰+47¹⁰+35¹⁰

See Computing Minimal Equal Sums Of Like Powers for detail.

Chen Shuwen noticed in 7 Sep1999 that the above result is also a solution of ( k = 2, 4, 6, 8, 10 ) .

[ 22, 61, 86, 127, 140, 151 ] = [ 35, 47, 94, 121, 146, 148 ]

By using Theorem 4, Chen Shuwen translated Nuutti Kuosa & Jean-Charles Meyrignac's result into an ideal solution of ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ) .

[ 0, 11, 24, 65, 90, 129, 173, 212, 237, 278, 291, 302 ] = [ 3, 5, 30, 57, 104, 116, 186, 198, 245, 272, 297, 299 ]

This is a great progress on the Prouhet-Tarry-Escott problem.

Last revised March,31, 2001.
Copyright 1997-2001, Chen Shuwen