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 8191 has a partner (Posted on 2018-07-26)
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 No Rating

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 computer solution | Comment 1 of 3
DefDbl A-Z
Dim crlf\$, n(999999, 5)

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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