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Three Power and Product Divisibility (Posted on 2015-09-19) Difficulty: 3 of 5
Find all pairs (X,Y) of positive integers such that each of 3X + 1 and 3Y + 1 is divisible by X*Y.
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
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)

 For tot = 2 To 62
   For x = 1 To tot / 2
      DoEvents
      y = tot - x
      If x < 32 And y < 32 Then
        v1 = Int(3 ^ x + 1.5)
        v2 = Int(3 ^ y + 1.5)
        d = x * y
        q1 = Int(v1 / d): r1 = v1 - q1 * d
        q2 = Int(v2 / d): r2 = v2 - q2 * d
        If r1 = 0 And r2 = 0 Then
           Text1.Text = Text1.Text & x & Str(y) & "      " & v1 & Str(v2)
           Text1.Text = Text1.Text & "   " & Str(d) & crlf
        End If
      End If
   Next
 Next


 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

finds only

1 1      4 4    1
1 2      4 10    2

i.e., (x,y) = (1,1) or (1,2), giving expression values of (4,4) or (4,10) respectively and products equal to 1 and 2.

Restrictions on x and y to at most 31 as 32 would risk rounding errors in the powers as going beyond the precision of double precision floating point.

  Posted by Charlie on 2015-09-20 07:56:20
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