Repdigit Base Finding A repdigit is a natural number that can be written solely by repeating the same digit. For example, 777 is a repdigit, since it's solely composed of the digit 7 repeated three times.
This isn't limited to simply decimal (base 10) numbers, however:

Every Mersenne number (of the form Mn = 2n-1) is a repdigit when written in binary (base 2).
Every number is trivially a repdigit when written in unary (base 1).
Every number n can also trivially be written as the repdigit 11 in base n-1 (for example, 17 when written in hexadecimal (base 16) is 11, and 3 when written in binary (base 2) is also 11).

The challenge here is to find other bases where the input number may be a repdigit.
Input
A positive integer x &gt; 3, in any convenient format.
Output
A positive integer b with (x-1) &gt; b &gt; 1 where the representation of x in base b is a repdigit.

If no such b exists, output 0 or some falsey value.
If multiple such b exist, you can output any or all of them.

Rules

The (x-1) &gt; b &gt; 1 restriction is to prevent the trivial conversions to unary or the &quot;subtract one&quot; base. The output number can be written in unary or any convenient base, but the base itself must not be one of the trivial conversions.
Input/output can be via any suitable method.
Standard loophole restrictions apply.

Examples
In --&gt; Out
11 --&gt; 0            (or other falsey value)
23 --&gt; 0            (or other falsey value)
55 --&gt; 10           (since 55 is 55 in base 10)
90 --&gt; 14           (since 90 is 66 in base 14 ... 17, 29, 44 also allowed)
91 --&gt; 9            (since 91 is 111 in base 9 ... 12 also allowed)
Jelly, 11 9 bytes

bRI¬P€TḊṖ


Returns a list of bases, which is empty (falsy) if there is none. Try it online!

How it works

bRI¬P€TḊṖ  Main link. Argument: x (number)

 R         Range; yield [1, ..., x].
b          Base; convert x to base n, for each n in the range.
  I        Compute the increment (differences of successive values) of each array
           of base-n digits. This yields only 0's for a repdigit.
   ¬       Apply logical NOT to each increment.
    P€     Compute the product of all lists of increments.
      T    Get the indices of all truthy products.
       Ḋ   Discard the first index (1).
        Ṗ  Discard the last index (x - 1).