Equal Sums of Like Powers

Non-negative Integer Solutions of

• 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 {a_i, b_i } <= 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 {a_i, b_i } <= 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 a_i and b_i 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 a_i and b_i 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
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