# Equal Sums of Like Powers

## Integer Solutions of

a1h + a2h + a3h = b1h + b2h + b3h
( h = 1, 2, 4 )
• A.Golden [5] gave the following theorem : ( Theorem 5 )
• If
[ a1 , a2 , ... , an+1 ] = [ b1 , b2 , ... , bn+1
is a non-symmetric solution of the system
a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k      ( k = 1, 2, ..., n
Then
[ Sa1 - T , Sa2 - T , ..., San+1 - T ] = [ Sb1 - T , Sb2 - T , ... , Sbn+1 - T ]
will be a non-trival integer solution of the system
a1h + a2h + ... + an+1h = b1h + b2h + ... + bn+1h      ( h = 1, 2, ..., n , n + 2 )
where T = a1+ a2 + ... + an+1 , S = n + 1 .
• Solution method of this type was first found in 1900's. ( See Dickson's book [9] )
• A.Golden [5] gave solutions of this type by starting with the non-symmetric solutions of the system ( k = 1, 2 ) .
• [ -7, 0, 7 ] = [ -8, 3, 5 ]
• [ -10, 1, 9 ] = [ -11, 5, 6 ]
• [ -12, 1, 11 ] = [ -13, 4, 9 ]
• [ -13, 0, 13 ] = [ -15, 7, 8 ]
• [ -48, 23, 25 ] = [ -47, 15, 32 ] = [ -45, 8, 37 ] = [ -43, 3, 40 ]

Last revised March,31, 2001.