`Equal Sums of Like Powers`

Non-negative Integer Solutions of

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

( k = 1, 2, 3, 4, 5, 6 )

The first solution of this type was found out by E.B.Escott in 1910. [2]

[ 0, 18, 27, 58, 64, 89, 101 ] = [ 1, 13, 38, 44, 75, 84, 102 ]

J.Chernick gave a two-parameter solution of this type in 1937. [7] Numerical examples by his method are

[ 0, 59, 68, 142, 181, 221, 267 ] = [ 1, 47, 87, 126, 200 , 209, 268 ]

No non-symmetric solution of this type is first found Chen Shuwen in 1997.

[ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]

This one is also the smallest solution of this type that have been found out.

[ 0, 14, 43, 141, 156, 193, 199 ] = [ 3, 9, 46, 133, 175, 176, 204 ]
[ 0, 24, 31, 74, 106, 137, 147 ] = [ 4, 11, 52, 57, 119, 126, 150 ]

By Chen Shuwen, in 2001.

Each non-symmetric solution has its "co-symmetric" solution., For example

[ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
[ 0, 14, 16, 45, 54, 73, 83 ] = [ 3, 5, 28, 34, 65, 66, 84 ]

In fact, these two solutions are equivalent.

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