Proposal for Brilliant Polymath Project

Classify all (rational) $q$ such that, if $e^q$ has the continued fraction expansion $[a_0;a_1,a_2,\ldots]$ , $\{ a_{mn+c} \}_{n=0}^\infty$ is an arithmetic progression for some $m$ and every $c>0$ .

(Conjectured: All $q$ rational have this property.)

As examples, observe the following expansions:

$\begin{aligned} q = 1, & e^q = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, \ldots], & m_{\min} = 3 \\ q = 2, & e^q = [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, \ldots], & m_{\min} = 5 \\ q = 1/2, & e^q = [1; 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, \ldots], & m_{\min} = 3 \\ \end{aligned}$

Known facts:

$e^z = \cfrac{1}{1 - \cfrac{z}{1 + z - \cfrac{z}{2 + z - \cfrac{2z}{3 + z - \cfrac{3z}{4 + z - \ddots}}}}}$

(Due to Euler.)

#NumberTheory

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Comments

How did you come across such observation?

Julian Poon - 5 years, 4 months ago

I noted the regularity of e's expansion in a YouTube lecture and thought that, since e is intimately linked with exponentiation, $e^x$ might display similar regularity as well.

Jake Lai - 5 years, 4 months ago

A good place to start might be to prove this for $q = \frac{1}{x}$ ( $x\in \mathbb{N}$ ), which Euler himself actually proved.

Eli Ross Staff - 5 years, 4 months ago

Where is this result found? It would be great to look at the proof for insight :)

Jake Lai - 5 years, 4 months ago

Here is an English translation of Euler's paper.

Eli Ross Staff - 5 years, 4 months ago

@Eli Ross – @Jake Lai Have you had a chance to think about this more? (I haven't, but am definitely curious.)

Eli Ross Staff - 5 years, 3 months ago

@Eli Ross – I have not, sadly. Been too busy with too much :<

Jake Lai - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$