@ Raptros_
What would be required to operationalize your thingummy? Assume I know nothing at all about what you’re up to at a technical level (and you’d be right). It might waste less of your time if I just listen in for a while at first …

Apropos of the discussion on computation, addition and subtraction would be miserable for sure, but multiplication and division would be easier, especially if (on paper) you lined up the terms or if, supposing you were to represent it as some object in the object-oriented-programming sense by an array of exponents of each particular prime. After all, you just add and subtract exponents.

Speaking of exponents, admin, it would be ever so much more tractable if instead of just concatenating, we had exponents. I had something like this in mind: 4 = 2×2 => 0^^ and 8 = 2x2x2 => 0^^^.

Of course, we could go another, more perverse, route and fix a positional rule such that a smaller number coming before a larger number implies that that larger is the exponent of the smaller. Something like 8 = 2^3 => 0(0). So when you have 7^6 (or something), you’d have to write (00) back to back 6 times.

Come to think of it, there are lots of places for that to fail. Compound exponents are also right out. Better scrap it.

Now, it’s telling (or is it?) that in order to do any real work, we have to go back to thinking in our mundane number system (that seems to be the point, actually). These are just some preliminary comments, I’ll try and formulate some more interesting thoughts later.

my code defines a lazy sequence of primes, using some decent techniques for filtering based on what’s already been computed. could be improved, of course, but frankly I don’t think anyone’s going to need to be able to evaluate expressions so far out that they’d notice the slowdown. (if anyone wants to take a shot at improving it, just send me a merge request on github if you come up with anything.)

the rest of the operations are defined on that sequence of primes.

[I’m especially interested because it should be impossible to code into Xeno without an ordinated matrix of the prime series (designating each prime first, second, third etc.), and the predominant part of modern cryptography relies on the fact that this matrix is irreducible to a formula — so I’m wondering how your program is working]

@admin
I’m working on an analysis of this. I’m not a number theorist, but this sort of thing is always interesting to me.