Equal Sums of Like Powers

Positive Integer Solutions of

• This type can be expressed as follows:
a₁ + a₂ + a₃ = b₁ + b₂ + b₃
1/a₁ + 1/a₂ + 1/a₃ = 1/b₁ + 1/b₂ + 1/b₃
• This type was first solved by Chen Shuwen in May of 2001.
• Numerical solutions can be obtained by computer easily. Here are some examples:
• [ 4, 10, 12 ] = [ 5, 6, 15 ]
• [ 6, 14, 14 ] = [ 7, 9, 18 ]
• [ 4, 9, 18 ] = [ 5, 6, 20 ]
• [ 5, 15, 15 ] = [ 7, 7, 21 ]
• [ 7, 14, 18 ] = [ 9, 9, 21 ]
• [ 2, 15, 20 ] = [ 3, 4, 30 ]
• [ 8, 20, 24 ] = [ 10, 12, 30 ]
• Chen Shuwen have checked at least 250 sets of solutions of this type.All of them satisfy the following relation:
a1 < b1 <= b2 < a2 <= a3 < b3
• See also Conjectures by Chen Shuwen.
• A sepecial case of the ( k = -1, 1 ) is:
a₁ + a₂ = b₁ + b₂ + b₃
1/a₁ + 1/a₂ = 1/b₁ + 1/b₂ + 1/b₃
• Chen Shuwen found the following solutions:
• [ 5, 45 ] = [ 8, 18, 24 ]
• [ 5, 56 ] = [ 7, 24, 30 ]
• [ 8, 72 ] = [ 15, 20, 45 ]
• [ 3, 40 ] = [ 4, 15, 24 ] = [ 5, 8, 30 ]
• [ 15, 168 ] = [ 21, 72, 90 ] = [ 28, 35, 120 ]
• It should be noticed that [ a₁, a₂ ] = [ b₁, b₂ , b₃ ] can not be denoted as [ 0, a₁, a₂ ] = [ b₁, b₂ , b₃ ].