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 1 is one of them (Posted on 2017-03-25)
There are 3 known numbers that are repunits at least in four bases.

1. The title mentions one of them.
2. Please find the other two without referring to Google, OEIS et al.
3. Limit your search up to 9999.
4. Looking for the "fourth" would be exercise in futility.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution | Comment 1 of 2
Every number is a repunit in base 1, so we need to find only three more bases.

`decimal   basesvalue     (written in decimal)31        2 5 308191      2 90 8190`

The last base shown in each case is also trivial (that is, in addition to base 1) as every number is a repunit (with representation 11) in the base that is one lower than the integer itself.

The results were found by

DefDbl A-Z
Dim crlf\$, n(9999, 6)

Form1.Visible = True

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

For bs = 2 To 9999
tot = 1
bp = bs
Do
tot = tot + bp
If tot < 10000 Then
n(tot, 0) = n(tot, 0) + 1
n(tot, n(tot, 0)) = bs
End If
bp = bp * bs
DoEvents
Loop Until tot > 9999
Next

For i = 2 To 9999
If n(i, 0) >= 3 Then
Text1.Text = Text1.Text & i
For j = 1 To n(i, 0)
Text1.Text = Text1.Text & Str(n(i, j))
Next j
Text1.Text = Text1.Text & crlf
DoEvents
End If
Next i

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

End Sub

 Posted by Charlie on 2017-03-25 14:51:18

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