# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k+ a5k = b1k + b2k + b3k + b4k + b5k
( k = 1, 3, 5, 7)
• A.Golden gave a method to solve this system in 1940's.   Numerical examples by his method are
• [ 3, 19, 37, 51, 53 ] = [ 9, 11, 43, 45, 55 ]
• [ 13, 31, 47, 65, 67 ] = [ 15, 27, 51, 61, 69 ]
• [ 1, 33, 39, 65, 71 ] = [ 11, 15, 53, 57, 73 ]
• [13, 53, 67, 103, 107 ] = [ 19, 39, 81, 93, 111 ]
• A.Letac found two solutions on the a1= 0 case of this type in 1942. 
• [ 0, 34, 58, 82, 98 ] = [ 13, 16, 69, 75, 99 ]
• [ 0, 63, 119, 161, 169 ] = [ 8, 50, 132, 148, 174 ]
• These two solutions lead to solutions of the ( k = 1, 2, 3, 4, 5, 6, 7, 8 ) type.
• Chen Shuwen proved by computer that there is no other solutions on the a1= 0 case of this type in the range of max { ai, bi }< 200.
• A.Moessner gave a two-parameter solution based on Golden's method in 1947.
• Chen Shuwen found a new method for this type in 1995. Here are some examples by his method. ( This solutions cannot be obtained by Golden's method.)
• [ 29, 75, 79, 125, 141 ] = [ 37, 53, 99, 117, 143 ]
• [ 85, 102, 110, 143, 145 ] = [ 90, 92, 117, 138, 148 ]
• [ 11, 79, 103, 149, 161 ] = [ 17, 63, 123, 135, 165 ]
• [ 17, 71, 103, 157, 163 ] = [ 31, 47, 121, 143, 169 ]

Last revised March,31, 2001.