Converting from m base to n base without arithmetic operations (Where m<n )

I have been working the last 8 years in try to convert from base n to base m without artihmetical operations.

My hypothesis is that by the same way that is possible convert bases without division (see the tables bellow.) based in a literal conversion you can create an equivalence table that allow this direct translation.

  00011111  |
(2)  |

00011111  | For convert in base 4 you take the follow number of digits.
x
2   = 4 | x=2 ( It is eaxctly)

[01][11][11]| You don't need a math operation for this operation because
The follow symbol '3'(4) -> '11'(2)
[1][3][3]|


 [0011][111]| You don't need a math operation for this operation because
The follow symbol '7'(8) -> '111'(2)
[3][7]|


As it is very know the number of groups that you have to use is:

Log (n) , where ( n can be represented as m^c , and c is and Integer) (m)

This operation:

'7'(8) -> '111'(2)   (Symbol(7)->111)


In my case, is a not math operations, because I have an state machine that is able to understand and reflect (bad joke) that symbol 7 means 111 in the default output (or default queue).

As you know when c is not an integer, we have a very complex problem therefore I was creating random table-states based on random rules (it means jumps of states using genetic algorithms) but It has been a real waste of time/energy.

Now I share my Idea, I believe that all bases must be represented as a sub-languages for other bases and they creates a cycle , It I couldn't demostrate it as a formal theorem. But my heart, my soul and my migth believes that it could be possible.

For example:

(3) 0  1  2  / 10  11  12 20 /
_______
3^1 (grp)  3^2(grp)


(2) 0  1  / 10  11  /
_______
2^1 (grp)  2^2(grp)

(4) 0  1 2 3 / 10 11 12 13 20 21 22 23 30 31 32 33 /
_______
4^1 (grp)  4^2(grp)

Conversion table:
GroupSize = Log(2)^4
Rule: (0/0  1/1 2/10  3/11 )
_______________________________________________________


Do you have any formalism to define a base as sub-language of other base for cases like this?

Post In Edition.

• (I find it hard to see base conversion when $m^k=n$.) Dec 3 '19 at 8:15
• Yes, I have the follow problems : The alphabets always are discret number but is normal that logs have a non integer result this is my big obstacle to create a unified m -> n base conversion. Dec 3 '19 at 13:50
• My other hypothesis is that exists a least common power, this is the philosophical stone, because is the size of the dictionary. Dec 3 '19 at 13:55
• My take: $log_m n \in \mathbb{N}$ trivial, $log_m n \in \mathbb{I}$ no finite grammar, no TM. Dec 3 '19 at 17:11
• This is part of the hypothesis, Sometimes I belive that it is as easy as synchronize 2 planets in the same point, each planet (base) after several cicles could to be in the same point in the same time again. Its like a least common multiple (LCM) but the cycles are exponentials. Dec 3 '19 at 19:44