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Linear*Quadratic = Square (Posted on 2016-11-12) Difficulty: 3 of 5
Find all integers x such that (1+x)(1+x2) is a perfect square.

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer exploration | Comment 1 of 2
Checking integers from -1,000,000 to 1,000,000 finds only

x  val sqrt
0    1   1
-1   0   0
1    4   2
7  400  20


DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For x0 = 0 To 1000000
  st = 2 * x0: fin = x0
  If st = 0 Then st = 1
  For x = -x0 To fin Step st
   DoEvents
   sq = (1 + x) * (1 + x * x)
   If sq >= 0 Then
    sr = Int(Sqr(sq) + 0.5)
    If sr * sr = sq Then
      Text1.Text = Text1.Text & x & Str(sq) & Str(sr) & crlf
    End If
   End If
  Next x
 Next x0
  
  
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


  Posted by Charlie on 2016-11-12 10:59:45
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