Open problems in continued fractions

[TeethWhitener]

TL;DR

New Nature paper by Raayoni et al. contains a number of new conjectures on continued fractions whose proofs might be within reach of the members of this forum

A paper out today in Nature might interest some folks in this forum:
https://www.nature.com/articles/s41586-021-03229-4
Permanent citation: Nature volume 590, pages67–73(2021)

The authors used machine learning to generate a large number of continued fraction expressions for fundamental constants such as $\pi$, $e$, $\zeta(3)$, and Catalan's constant. Several of these expressions were already known, but many are still unproven. The website for the project is here:
http://www.ramanujanmachine.com/
and they've posted many of the open problems here:
http://www.ramanujanmachine.com/results/
The supplementary info for the Nature article gives a few proofs of (formerly) open conjectures, and many of these seem well within the ability of a sharp undergrad--though still well outside my own abilities. In some cases, the proofs employ nothing more complicated than finding the right special function identities in a math handbook. I've seen how talented some of this forum's members are, so I thought many of you might be interested at taking a crack at some of these open conjectures. Best of luck!

 

[jedishrfu]

This is nice. I really like the discovery and team idea behind the website although what usually happens is that multiple people discover the same thing and then it's the first one to get in there and the others get discouraged as they get beat to the punch each time. (aka wack-a-mole)

Perhaps a better course would be to have a contest with the discoverers only selecting the best name for the newly discovered identity.

I once thought of a math game where one student would use known identities to create a complex identity and for others to prove it. Of course, there are two sides to the coin in that mistakes made in creating it will cause issues in reducing it down but programming can fix anything (said with a wonky smile) :-?

 

[Keith_McClary]

The pattern of some of these is not obvious to me:

 

[TeethWhitener]

Those might have been odd men out. For most of the conjectures on the site, they give two polynomials $a_n$ and $b_n$ such that the continued fraction is given by
$$a_0 + \frac{b_1}{a_1+\frac{b_2}{a_2+\frac{b_3}{a_3+\cdots}}}$$
E.g. http://www.ramanujanmachine.com/wp-content/uploads/2020/06/zeta3.pdf

 

[jedishrfu]

How do you attack these types of fractions to prove them?

Do you start with a known solution and then do some magic on both sides to transform them into the one you’re proving?

or do you have to go back to the $a_n$ and $b_n$ generating expressions?

 

[fresh_42]

How do you attack these types of fractions to prove them?

There are normally repetitions which allow a recursion formula. My first thought was:
If Euler didn't mention them, then they are either too complicated or not interesting at all.

 

[dromarand]

$\frac{2}{ \pi } = 1\cdot \frac{1}{2\cdot \frac{2}{3\cdot \frac{3}{\ddots} } }$