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Cube root sum equality puzzle (Posted on 2015-06-09) Difficulty: 3 of 5
Find all pairs (a,b) of positive integers that satisfy this equation:

(∛A + ∛B -1)2 = 49 + 20*∛6

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
DefDbl A-Z
Dim crlf$

Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 rhs = 49 + 20 * 6 ^ (1 / 3)
 sr = Sqr(rhs)
 
 For a = 1 To 1100
  b = Int((sr + 1 - a ^ (1 / 3)) ^ 3 + 0.5)
  If b > 0 Then
    lhs = (a ^ (1 / 3) + b ^ (1 / 3) - 1) ^ 2
    diff = Abs(rhs - lhs)
    If diff / rhs < 0.000001 Then
      Text1.Text = Text1.Text & a & Str(b) & "    "
      Text1.Text = Text1.Text & lhs & Str(rhs) & crlf
      DoEvents
    End If
  End If
 Next
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

finds

A   B            LHS            RHS  
48 288    85.3424118566428 85.3424118566428
288 48    85.3424118566428 85.3424118566428

so the answer is

(48,288) and its reverse, (288,48).


  Posted by Charlie on 2015-06-09 15:54:15
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