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?

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