# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k = b1k + b2k + b3k
( k = 1, 3 )
• A.Moessner gave some parameter solutions of this type in 1939. Numerical exampls are
• [ 0, 7, 8 ] = [ 1, 5, 9 ]
• [ 0, 7, 9 ] = [ 2, 4, 10 ]
• [ 12, 23, 28 ] = [ 13, 21, 29 ]
• Smallest solutions of this type can be found easily by using computer.
• [ 1, 5, 5 ] = [ 2, 3, 6 ]
• [ 0, 7, 8 ] = [ 1, 5, 9 ]
• [ 0, 7, 9 ] = [ 2, 4, 10 ]
• [ 1, 9, 9 ] = [ 4, 4, 11 ]
• [ 2, 8, 10 ] = [ 4, 5, 11 ]
• [ 5, 10, 11 ] = [ 6, 8, 12 ]
• Solution chain of this type had been studied by A.Choudhry in 1992. However, solutions by his method are in integers, not in positive integers.
• The positive integer solution chain is first obtained by Chen Shuwen in 1995.
• [ 2, 52, 58 ] = [ 4, 46, 62 ] = [ 13, 32, 67 ] = [ 22, 22, 68 ]
• [ 5, 58, 70 ] = [ 7, 53, 73 ] = [ 13, 43, 77 ] = [ 21, 33, 79 ]
• Jarek Wroblewski find the following results by exhaustive computer search in 2001:
• [ 10, 214, 215 ] = [ 19, 179, 241 ] = [ 43, 139, 257 ] = [ 49, 131, 259 ] = [87, 88, 264]
• [ 89, 361, 367 ] = [ 97, 323, 397 ] = [ 115, 285, 417 ] = [ 133, 257, 427 ] = [ 152, 232, 433 ] = [ 187, 193, 437 ]
• [ 16, 624, 699 ] = [ 19, 611, 709 ] = [ 39, 555, 745 ] = [ 79, 481, 779 ] = [ 107, 439, 793 ] = [ 169, 359, 811 ] = [ 187, 338, 814 ] = [ 259, 261, 819 ]
• A solution chain of length j=65, with 87-88 digit numbers, is also found by Jarek Wroblewski in 2001.

Last revised Oct, 16, 2001.