Can someone tldr this please?
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.
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"