### The Physics Examples...

One of the most interesting help page about the use of the Physics package is Physics,Examples. This page received some additions recently. It is also an excellent example of the File -> Export -> LaTeX capabilities under development.

Below you see the sections and subsections of this page. At the bottom, you have links to the updated PhysicsExample.mw worksheet, together with PhysicsExamples.PDF.

The PDF file has 74 pages and is obtained by going File -> Export -> LaTeX (FEL) on this worksheet to get a .tex version of it using an experimental version of Maple under development. The .tex file that results from FEL (used to get the PDF using TexShop on a Mac) has no manual editing. This illustrates new automatic line-breakingequation labels, colours, plots, and the new LaTeX translation of sophisticated mathematical physics notation used in the Physics package (command Latex in the Maplesoft Physics Updates, to be renamed as latex in the upcoming Maple release).

In brief, this LaTeX project aims at writing entire course lessons or scientific papers directly in the Maple worksheet that combines what-you-see-is-what-you-get editing capabilities with the Maple computational engine to produce mathematical results. And from there get a LaTeX version of the work in two clicks, optionally hiding all the input (View -> Show/Hide -> Input).

PS: MANY THANKS to all of you who provided so-valuable feedback on the new Latex here in Mapleprimes.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

### An elementary problem

Here is a very nice (but not easy) elementary problem.
The equality
ceil(2/(2^(1/n)-1)) = floor(2*n/ln(2));

is not an identity, it does not hold for each positive integer n.
How to find such a number?

[This is a re-post, because the original vanished when trying a conversion Question-->Post]

The problem appears in the recent book:
Richard P. Stanley - Conversational Problem Solving. AMS, 2020.

The problem is related to a n-dimensional tic-tac-toe game. The first counterexample (2000) was wrong due to a multiprecision arithmetic error.
The  author of the book writes
"To my knowledge, only eight values of n are known for which the equation fails,
and it is not known whether there are infinitely many such values",

but using Maple it will be easy to find more.

A brute-force solution is problematic because the smallest counterexample is > 7*10^14.

```restart;
a := 2/(2^(1/n)-1): b := 2*n/ln(2):
asympt(b-a, n);
```

It results:  b - a → 1 (for n →oo);
So, to have a counterexample, b must be close to an integer
m ≈ 2*n/ln(2)  ==> n/m ≈ ln(2)/2

The candidates for n/m will be obviously the convergents of the continued fraction of the irrational number ln(2)/2.

```convert(ln(2)/2, confrac, 200, 'L'):
Digits:=500:
for n in numer~(L[3..]) do
if not evalf(ceil(a)=floor(b)) then printf("n=%d\n", n) fi;
od:
```

n=777451915729368
n=140894092055857794
n=1526223088619171207
n=54545811706258836911039145
n=624965662836733496131286135873807507
n=1667672249427111806462471627630318921648499
n=36465374036664559522628534720215805439659141
n=2424113537313479652351566323080535902276508627
n=123447463532804139472316739803506251988903644272
n=97841697218028095572510076719589816668243339678931971
n=5630139432241886550932967438485653485900841911029964871
n=678285039039320287244063811222441860326049085269592368999
n=312248823968901304612135523777926467950572570270886324722782642817828920779530446911
n=5126378297284476009502432193466392279080801593096986305822277185206388903158084832387
n=1868266384496708840693682923003493054768730136715216748598418855972395912786276854715767
n=726011811269636138610858097839553470902342131901683076550627061487326331082639308139922553824778693815

So, we have obtained 16 counterexamples. The question whether there are an infinity of such n's remains open.

## Plotting in a plane

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