Equal Sums of Like Powers

Integer Solutions of

• A.Golden [5] gave the following theorem : ( Theorem 5 )
If
[ a₁ , a₂ , ... , a_n+1 ] = [ b₁ , b₂ , ... , b_n+1 ] 
is a non-symmetric solution of the system
a₁^k + a₂^k + ... + a_n+1^k = b₁^k + b₂^k + ... + b_n+1^k      ( k = 1, 2, ..., n ) 
Then
 [ Sa₁ - T , Sa₂ - T , ..., Sa_n+1 - T ] = [ Sb₁ - T , Sb₂ - T , ... , Sb_n+1 - T ]
will be a non-trival integer solution of the system
a₁^h + a₂^h + ... + a_n+1^h = b₁^h + b₂^h + ... + b_n+1^h      ( h = 1, 2, ..., n , n + 2 ) 
where T = a₁+ a₂ + ... + a_n+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 ]