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Quadratic and Perfect Square Query (Posted on 2016-04-30) Difficulty: 3 of 5
Determine all possible pairs (X,Y) of positive integers such that each of
X2 +Y and Y2 + 17*X is a perfect square.

No Solution Yet Submitted by K Sengupta    
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Solution computer exploration -- probable spoiler | Comment 1 of 2
Each line below shows x, y, the values of the two formulae, and their square roots.

1 8     9 81      3 9
3 7     16 100      4 10
24 49     625 2809      25 53
20 84     484 7396      22 86
33 280     1369 78961      37 281

based on the following program, manually stopped at about x + y = 100,000 so all totals of x+y under that value are covered:

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 2 To 999999
   For x = 1 To tot - 1
    DoEvents
     y = tot - x
     lhs = x * x + y
     sr1 = Int(Sqr(lhs) + 0.5)
     If sr1 * sr1 = lhs Then
       rhs = y * y + 17 * x
       sr = Int(Sqr(rhs) + 0.5)
       If sr * sr = rhs Then
         Text1.Text = Text1.Text & x & Str(y) & "     " & lhs & Str(rhs) & "      "
         Text1.Text = Text1.Text & sr1 & Str(sr) & crlf
       End If
     End If
   Next
 Next
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


  Posted by Charlie on 2016-04-30 09:05:23
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