According to the Wikipedia page on generalized continued fractions, $\pi$ can be given several GCF representations which have very regular structures; for example, one has the partial denominators as (1, 2, 2, 2, …) and the partial numerators as (4, 12, 32, 52, 72, …). My question is, can every real number be represented as a GCF that exhibits some sort of "structure"? I put that word in quotes because I'm not really sure how to define it concretely; ideally, the sequence of either partial numerators or partial denominators should be expressable using a predictable, explicit formula. Are there any theorems establishing results like this, or are there some numbers that have no GCF representation that is indistinguishable from random noise? If some numbers need to be expressed in other forms for a pattern to emerge, like Engel expansions, I'd be interested in knowing about that too.
[Math] Patterns in Generalized Continued Fractions
nt.number-theory
Best Answer
I think under any reasonable definition there will be only countably many explicit formulas or patterns. So in fact most reals can't be expressed this way. (See also the concept of "computable number.")