I am fascinated by the fact that "important" irrational numbers like the golden ratio, base of the natural exponent, pi, square roots have a "regular" representation as an infinite continued fraction.
Is there some correspondence between "having a regular ICF representation" and the "importance" of a number, whatever the meaning of "importance" might be? Is this a thing, or is such connection spurious and you could find a "regular" representation for any irrational number if you wanted to?
The opposite question also interests me: if I choose some "regular" representation like [1;1,2,3,2,3,4,3,4,5,4,5,6,...] what are the chances for it's not going to be a square root of some number, for example?