I was reading about Godel Numbering I was wondering if it is possible to create a unique number in a way (i.e. as a product), but with that the requirement the number does not grow at the rate of the numbers being multiplied.
Example for simplicity assume we map $d$ to $5$ and $g$ to $7$
The following sequence: $ddddddddddg = 5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot7$
and the following sequence: $ggggggggggd$ = $7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot5$
generates different products.
Is there a way to generate unique and different products but the resulting product is a lower number than the one the approach above produces?
I thought the following:
Use the unique characters and multiply their occurrences i.e.
$ddddddddddg => d10g1 => d\cdot10\cdot g\cdot1$
but due to associativity that is the same as:
$ggggggggggd => g10d1 => g\cdot10\cdot d\cdot1$
Is there some variation of Godel numbering that creates smaller products?