3 ms·

There are a lot of continued fraction formulas for mathematical constants of the form: c = a(0) + a(1) / (b(1) + a(2) / ( .... Here a and b are polynomials (t

There are a lot of continued fraction formulas for mathematical constants of the form: c = a(0) + a(1) / (b(1) + a(2) / ( .... Here a and b are polynomials (this looks a bit more sensible in their paper, I don't really know how to denote continued fractions in ASCII). If you stop this fraction at a certain point you get a fraction like p_n / q_n. In general the value of p_n and q_n grow extremely rapidly. Their conjecture is that if you look at this fraction in its lowest terms then p_n and q_n grow way less rapidly (exponential vs. a power of a factorial), but only if the limit is a mathematical constant. Obviously this can't hold for all mathematical constants, but as a rule of thumb it seems to work well enough to identify when two polynomials a and b correspond to some mathematical constant or a combination of mathematical constants.

Garlef 3y ago
And what is a "mathematical constant" in this context? Not in terms of examples but rather: Is this something that can be defined in a meaningful way? Remembering that one joke about inductive proofs ("there are no uninteresting natural numbers"), my first impression would be that this is a bit of a void concept. Regarding the joke: "there are no uninteresting natural numbers because if there were any, there would also be a smallest uninteresting number, making this number interesting, which is a contradiction"
- jjgreen 3y ago
  I'd say not, mathematical constants are usually irrational (often transcendental) numbers which tend to crop-up consistently in different contexts, so they're culturally (and so ill-) defined. Such concepts are not uncommon in mathematics, "canonical" springs to mind.