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8191 has a partner (Posted on 2018-07-26) Difficulty: 3 of 5
8191 is a more-than-two-digit repunit prime in two bases:
8191 = 111 in base 90 = 1111111111111 in base 2.

Find the only other number with this property.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution | Comment 1 of 3
DefDbl A-Z
Dim crlf$, n(999999, 5)
Private Sub Form_Load()

crlf = Chr(13) + Chr(10)
Text1.Text = ""

 For b = 2 To 500
 For p = 3 To 100
   v = Int(b ^ p - 0.5) / (b - 1)
   If v > 999999 Then Exit For
  ' Text1.Text = Text1.Text & v & crlf
   n(v, 0) = n(v, 0) + 1
   n(v, n(v, 0)) = b
   DoEvents
 Next
 Next
 
 For i = 1 To 999999
   If n(i, 0) > 1 Then
      Text1.Text = Text1.Text & i & Str(n(i, 0)) & crlf
      For j = 1 To n(i, 0)
        Text1.Text = Text1.Text & Str(n(i, j))
      Next
      Text1.Text = Text1.Text & crlf
   End If
 Next
 
 Text1.Text = Text1.Text & crlf & tot & " done"
  
End Sub

finds

31 2
 2 5
8191 2
 2 90
 
meaning 31 has 2 repunit representations: base 2 and base 5 
    and 8191 has 2 repunit representations: base 2 and base 90
 
The answer is 31, which is a repunit in base 2 and in base 5. 

Edited on July 26, 2018, 12:20 pm
  Posted by Charlie on 2018-07-26 11:26:45

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