# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k = b1k + b2k + b3k
( k = 2, 4 )
• This system had been considered since 1900's.[9]
• A.Golden gave some parameter solutions of this system in 1940's. [5] [38] [20]
• Solutions of this system can be seperated into two kind. One kind also satisfy a1+a2=a3 and b1+ b2= b3.Solutions of this kind are easy to obtained.
• [ 0, 7, 7 ] = [ 3, 5, 8 ]
• [ 1, 9, 10 ] = [ 5, 6, 11 ]
• [ 1, 11, 12 ] = [ 4, 9, 13 ]
• [ 2, 11, 13 ] = [ 7, 7, 14 ]
• [ 0, 13, 13 ] = [ 7, 8, 15 ]
• The other kind of solutions does not satisfy a1+a2=a3 and b1+ b2= b3. Solutions of this kind may lead to solutions of the type ( k = 1, 3, 5, 7 ), ( k = 2, 4, 6, 8 ), ( k = 1, 2, 3, 4, 5, 6, 7, 8 ) and ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 ).
• [ 9, 25, 29 ] = [ 15, 19, 31 ]
• [ 7, 30, 31 ] = [ 18, 19, 35 ]
• [ 12, 29, 35 ] = [ 20, 21, 37 ]
• [ 6, 25, 37 ] = [ 15, 19, 38 ]
• [ 5, 32, 41 ] = [ 16, 25, 43 ]
• [ 2, 29, 45 ] = [ 15, 23, 46 ]
• [ 1, 30, 49 ] = [ 19, 21, 50 ]
• Method for solution Chain of this system was obtained by A.Golden in 1940's.[5] [3]
• [ 28, 175, 203 ] = [ 77, 140, 217 ] = [ 107, 113, 220 ] = [ 5, 188, 193 ] = [ 67, 148, 215 ] = [ 55, 157, 212 ]

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