# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k+ a5k+ a6k = b1k + b2k + b3k + b4k + b5k + b6k
( k = 1, 2, 3, 4, 5 )
• G.Tarry gave a two-parameter solution of this type in 1912.[23] Numerical examples are
• [ 0, 3, 5, 11, 13, 16 ] = [ 1, 1, 8, 8, 15, 15 ]
• [ 0, 5, 6, 16, 17, 22 ] = [ 1, 2, 10, 12, 20, 21 ]
• A.Golden gave a four-parameter solution in 1912.[5]
• Only one non-symmetric solution of this type was appeared in the literature.[5] ( Page27 ) It's obtained by A.Golden. However, Golden's method for this non-symmetric solution was unknown.
• [ 0, 19, 25, 57, 62, 86 ] = [ 2, 11, 40, 42, 69, 85 ]
• Chen Shuwen found how to get non-symmetric solution of this type in 1995. Here are some examples by his method:
• [ 0, 9, 17, 34, 36, 46 ] = [ 1, 6, 24, 25, 42, 44 ]
• [ 0, 6, 23, 38, 47, 57 ] = [ 2, 3, 27, 33, 50, 56 ]
• [ 0, 8, 27, 45, 46, 61 ] = [ 1, 6, 33, 35, 52, 60 ]
• [ 0, 14, 17, 46, 51, 67 ] = [ 2, 7, 25, 39, 56, 66 ]
• A.Golden gave a parameter method for solution chain of this type in 1940's.[5] ( Page90 ) Numerical example is
•    [ 0, 567, 644, 1778, 1855, 2422 ]
• = [ 2, 535, 678, 1744, 1887, 2420 ]
= [ 7, 490, 728, 1694, 1932, 2415 ]
= [ 15, 444, 782, 1640, 1978, 2407 ]
= [ 28, 392, 847, 1575, 2030, 2394 ]
= [ 42, 350, 903, 1519, 2072, 2380 ]
= [ 62, 303, 970, 1452, 2119, 2360 ]
= [ 70, 287, 994, 1428, 2135, 2352 ]
= [ 95, 244, 1062, 1360, 2178, 2327 ]
= [ 103, 232, 1082, 1340, 2190, 2319 ]
= [ 119, 210, 1120, 1302, 2212, 2303 ]
= [ 144, 180, 1175, 1247, 2242, 2278 ]

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