# 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, 6 )
• Chen Shuwen Studied this type since 1995. He found a method to solve this system and obtained the following examples.
• [ 116, 166, 206, 331, 336, 411 ] = [ 131, 136, 236, 291, 366, 406 ]
• This one is the first solution obtained by Chen in 1995. ( It took a 386SX/33 PC run one night.)
• [ 11, 23, 24, 47, 64, 70 ] = [ 14, 15, 31, 44, 67, 68 ]
• [ 42, 48, 59, 74, 76, 85 ] = [ 43, 46, 62, 69, 80, 84 ]
• [ 23, 31, 60, 80, 91, 103 ] = [ 25, 28, 65, 73, 96, 101 ]
• [ 19 , 29 , 89 , 123 , 127 , 152 ] = [ 23 , 24 , 97 , 107 , 139 , 149 ]
• [ 73 , 85 , 116 , 146 , 147 , 164 ] = [ 74 , 83 , 120 , 136 , 157 , 161 ]
• By computer search, Chen Shuwen (2017) find that there are only 8 sets of solutions in the range of max {ai, bi } <= 100.
• [ 1, 14, 17, 46, 48, 60 ] = [ 4, 6, 25, 38, 56, 57 ]
• [ 10, 21, 25, 44, 62, 68 ] = [ 13, 14, 32, 40, 65, 66 ]
• [ 11, 23, 24, 47, 64, 70 ] = [ 14, 15, 31, 44, 67, 68 ]
• [ 7, 19, 30, 50, 68, 76 ] = [ 8, 16, 34, 47, 70, 75 ]
• [ 42, 48, 59, 74, 76, 85 ] = [ 43, 46, 62, 69, 80, 84 ]
• [ 3, 22, 28, 74, 76, 95 ] = [ 8, 10, 39, 62, 88, 91 ]
• [ 3, 22, 28, 59, 76, 95 ] = [ 7, 11, 43, 48, 80, 94 ]
• [ 1, 16, 33, 69, 70, 97 ] = [ 6, 7, 43, 55, 79, 96 ]
• By computer search, Chen Shuwen (2017) prove that there is no prime solution of this system in the range of max {ai, bi } <= 3600.
• A.Choudhry find the below parametric solution for ( k = 0, 1, 2, 3, 4 ) in 2016. [1602]
• a1 = ( d - 30 f 2 + 4 f g + 2 g 2 ) (d + 6 f 2 + 12 f g + 6 g 2 )
• a2 = ( d + 30 f 2 - 4 f g - 2 g 2 ) (d - 6 f 2 + 4 f g + 10 g 2 )
• a3 = ( d - 18 f 2 + 12 f g - 2 g 2 ) (d + 18 f 2 + 20 f g + 2 g 2 )
• a4 = ( d + 6 f 2 - 4 f g - 10 g 2 ) (d + 18 f 2 - 12 f g + 2 g 2 )
• a5 = ( d - 6 f 2 + 20 f g - 6 g 2 ) (d - 18 f 2 - 20 f g - 2 g 2 )
• a6 = ( d - 6 f 2 - 12 f g - 6 g 2 ) (d + 6 f 2 - 20 f g + 6 g 2 )
• b1 = ( d - 18 f 2 + 12 f g - 2 g 2 ) (d - 6 f 2 + 4 f g + 10 g 2 )
• b2 = ( d + 18 f 2 + 20 f g + 2 g 2 ) (d + 6 f 2 - 20 f g + 6 g 2 )
• b3 = ( d - 6 f 2 + 20 f g - 6 g 2 ) (d + 6 f 2 + 12 f g + 6 g 2 )
• b4 = ( d - 6 f 2 - 12 f g - 6 g 2 ) (d + 30 f 2 - 4 f g - 2 g 2 )
• b5 = ( d + 6 f 2 - 4 f g - 10 g 2 ) (d - 30 f 2 + 4 f g + 2 g 2 )
• b6 = ( d - 18 f 2 - 20 f g - 2 g 2 ) (d + 18 f 2 - 12 f g + 2 g 2 )
• Chen Shuwen notice in 2017 that, for any f and g, when d is large enough, all ai and bi will be positive integers. That is to say, Choudhry's above parametric solution can lead to ideal non-negative integer solution of ( k = 0, 1, 2, 3, 4 ) . Fox example, taking f = 2, g = 1, d = 180, then we have
• [ 209, 273, 357, 637, 713, 841 ] = [ 217, 253, 377, 609, 741, 833 ]   ( k = 0, 1, 2, 3, 4 )
• Based on Choudhry's above parametric solution for ( k = 0, 1, 2, 3, 4 ), Chen Shuwen (2017) obtain the below parametric solution for ( k = 1, 2, 3, 4, 6 ).
• a1 = d2 + 40 d ( 3 f 2 - 2 f g - g 2 ) + 4 ( f + g )2 ( 99 f 2 + 54 f g - 29 g 2 )
• a2 = d2 - 40 d ( 3 f 2 + g 2 ) + 4 ( 99 f 4 - 348 f 3 g - 342 f 2 g 2 + 116 f g3 + 11 g4 )
• a3 = d2 - 160 d f g + 4 ( - 3 f + g )2 ( 31 f 2 + 22 f g - 9 g 2 )
• a4 = d2 - 40 d ( 3 f 2 - 2 f g - g 2 ) - 4 ( - 3 f + g )2 (29 f 2 + 18 f g - 11 g 2 )
• a5 = d2 + 40 d ( 3 f 2 + g 2 ) - 4 ( 261 f 4 - 132 f 3 g - 378 f 2 g 2 + 44 f g3 + 29 g4 )
• a6 = d2 + 160 d f g - 4 ( f + g )2 ( 81 f 2 + 66 f g - 31 g 2 )
• b1 = d2 + 40 d ( 3 f 2 - 2 f g - g 2 )- 4 ( - 3 f + g )2 ( 29 f 2 + 18 f g - 11 g 2 )
• b2 = d2 - 40 d ( 3 f 2 + g 2 ) - 4 ( 261 f 4 - 132 f 3 g - 378 f 2 g 2 + 44 f g3 + 29 g4 )
• b3 = d2 - 160 d f g - 4 ( f + g )2 ( 81 f 2 + 66 f g - 31 g 2 )
• b4 = d2 - 40 d ( 3 f 2 - 2 f g - g 2 ) + 4 ( f + g )2 ( 99 f 2 + 54 f g - 29 g 2 )
• b5 = d2 + 40 d ( 3 f 2 + g 2 ) + 4 ( 99 f 4 - 348 f 3 g - 342 f 2 g 2 + 116 f g3 + 11 g4 )
• b6 = d2 + 160 d f g + 4 ( - 3 f + g)2 ( 31 f 2 + 22 f g - 9 g 2 )
• Chen Shuwen notice that, for any f and g, when d is large enough, all ai and bi will be positive integers. That means this parametric solution can lead to ideal non-negative integer solution of ( k = 1, 2, 3, 4, 6 ) . Fox example, taking f = 2, g = 1, d = 540, we have
• [ 5, 421, 449, 1497, 1533, 1885 ] = [ 13, 345, 525, 1429, 1601, 1877 ]   ( k = 1, 2, 3, 4, 6 )

Last revised Mar 19, 2017
Copyright 1997-2017, Chen Shuwen