exponents

I tried it with the birthday problem[1] (real answer 23) and got 16 if I used whole exponents[2] and 19 with fractional exponents[3].

Nature is still free to define huge exponents to difficult problems, making them essentially unsolvable in practice.

Right - by fitting the pattern but failing the actual criterion, it disproves the consecutive exponents hypothesis.

It parses arbitrary negative exponents fine, but it does not recognize that 1e-324 is greater than 0.

That's the sort of arithmetic that you'll want to keep in your head, the mathematics of exponents.

You can easily extend prime factorization to rational numbers by allowing negative as well as nonnegative exponents for primes.

Our notation for exponents is okay but not great, and having two notations for the exponential function is especially confusing.

First it can interpret certain string-patterns as numbers with exponents, and secondly zero raised to any power is always zero.

Correct me if I'm wrong, but aren't you supposed to evaluate stacked exponents from the top down?

Halving isn't really a big issue, when we're talking about something that normally talk about in exponents.

Is deleting all the exponents on both sides a valid change as long as I do it to both sides?

I was not aware that they didn't support fractions in exponents; I don't think I had ever tried them for that.

Maybe the criteria and exponents have been arbitrarily chosen to "get the right answers" according to someone's intuitive notion of similarity.

"In practice we find that algorithms in P usually have small exponents." We find that algorithms we have identified in P usually have small exponents.

I've watched over his shoulder as he gets to 8192 in a game of 2048 (original rules, no "undo".) He understands exponents and logarithms.

So in both cases it becomes just, "what's the configuration, what symmetries does it have" that determines how macroscopic parameters (whether critical exponents or eigenvalue densities) ultimately behave.