Does some mathematician knows if the above fulguration I had is a known fact?

Yeah, fortunately! That succession of powers is the base of the positional notation used to represent numbers!

So, what do you get when you subtract 1 to a power or 10? a string of nines. What do you get when you do the same to a power of 16 in hexadecimal? a string of F's. What do you get when you subtract one to a power of 2 in binary? a string of ones.

For instance:

  9000  = 9 * 1000 = (10 - 1) * 10 ** 3
   900  = 9 *  100 = (10 - 1) * 10 ** 2
    90  = 9 *   10 = (10 - 1) * 10 ** 1
 +   9  = 9 *    1 = (10 - 1) * 10 ** 0
 -----
  9999  = 9 * 1111 = (10 - 1) * 1111

  9999
 +   1
 -----
 10000  = 10 ** 4
[download]