## Monday, September 14, 2020

### An interesting serendipitous number

Last seek I blogged about two math problems of interest to me here.

One of them two people posted answers, which was great since I didn't know how to solve them and now I do. Yeah! I blogged about that here.

The other problem got no comments, so I suppose it was of interest to me but not others. I was interested in it because the story behind it is interesting, and the answer is interesting.

it is from the paper

An interesting and serendipitous number by John Ewing and Ciprian Foias, which is a chapter in the wonderful book

Finite vs Infinite: Contributions to an eternal dilemma

Here is the story, I paraphrase the article (I'll give pointers  later).

In the mid 1970's a student asked Ciprian about the following math-competition problem:

x(1)>0    x(n+1) =  (1 + (1/x(n)))^n. For which x(1) does x(n) --> infinity?

It turned out this was a misprint. The actual problem was

x(1)>0  x(n+1)=(1+(1/x(n))^{x(n)}. For which x(1) does x(n) --> infinity.

The actual math-comp problem  (with exp x(n)) is fairly easy (I leave it to you.) But this left the misprinted problem (with exp n).  Crispian proved that there is exactly ONE x(1) such that x(n)--> infinity.

Its approx 1.187... and may be trans.

I find the story and the result interesting, but the proof is to long for a blog post.

I tried to find the article online and could not. A colleague found the following:

A preview of the start of the article here

Mathworld page on that number here

Most of the article but skips two pages here

1. The article is easily found... For instance https://doi.org/10.1007/978-1-4471-0751-4_8 . Of course, it is behind a paywall, but then Sci-hub comes to rescue and you find the complete book: http://libgen.rs/book/index.php?md5=231BF372E8C89E575F5A4A0C89D687A6

2. This reminded me of how, as n -> Inf, (1 + 1/n)^n is about e. So I could imagine that, for "large enough" n, if x_n is "a bit more than" n, then 1/x_n is "a bit less than" 1/n, and x_{n+1} is "somewhat less than" e. Which means that x_{n+2} will have something^{n+2}, which seemed apt to be even larger. So it seemed like it should oscillate, at least sometimes. (Also, induction shows that x_n >= 1 always.)

At that point, I basically punted: by writing the five lines of Python to see that this happened (at least sometimes), and looking at the solution in this post :/

3. Suddenly invented the following