Represent large integers with exponents.

Question

I have recently become interested in somewhat large numbers. What I would like to know is if there is a way to express these numbers with exact precision using only exponents. For example I was watching a Numberphile video on youtube that talks about graham's number and in the explanation they state 3^3^3 is 7,625,597,484,987. Can any number be expressed with a similar process. If so how would one go about expressing the number 8,947,996,536,373,196 in that fashion.

Answer 1

Not only is this doomed to failure, it's doomed to failure in a very broad sense and for a subtle but deep reason: because you can't stuff ten pounds of apples into a four-pound bag.

Imagine that every number had a 'description' that was smaller than it was (we'll be more specific about description later). Then nothing prevents you, in principle, from treating the description itself as a number, and applying the same principle to it, to get a smaller number — and continuing this process, all the way down. But you can't assign a unique one-digit number to every hundred-digit number, for instance - you can't even assign a unique 99-digit number to every hundred-digit number, so you can't expect to shrink every number.

This may all seem obvious, but it has a formalization; the concept of Kolmogorov Complexity, which talks about the shortest 'program' (on a universal computer) to compute a number. And a more formal version of this argument can be used to show that not only do there have to be numbers that have no simpler description than just themselves, but in fact that almost all numbers (in a formalizable sense) have essentially no simpler description than just writing out the number. Numbers like $3^{3^{3^3}}$ or even $5^{5^5}-125^2+1$ — or any number that we can simplify — are the exception, not the rule.

Answer 2

I think you want prime number decomposition (plus maybe some further simplification in the exponents, if applicable - in the case of 3^3^3, you recognize 27=3^3). Any number has a prime decomposition, but if it's prime, it's already in that form (nothing happens). Only a very few specially chosen numbers have a short and simple form - those, that have a special rule/symmetry/pattern about them, which you can write down in some form.

You can understand this without math, actually. It's a simple matter of amount of information you need to tell the number. Start with your 3^3^3... and imagine how you would tell all the integers smaller than it. Of course... there are 3^3^3 of them, and there's no way of representing all of them in a form shorter than the original, because there are fewer combinations of digits and operators than numbers you want to specify. That's the main property of data compression in general (zip, for instance). Files are essentially just very long binary numbers. Those that have some patterns and rules, can be compressed, but in general, you'll run out of symbols/characters/digits if you wanted all of them to get shorter.

Answer 3

For any integer, $N$, we can write :

$$ N = p_1^a . p_2^b . p_3^c . . . . $$

where $p_i$ are the prime factors of N.

Further levels of exponents depends on the factorisability of the primary exponents, $a, b, c, ...$