Novel examples, proofs or results in mathematics from arithmetic billiards

Question

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….

Wikipedia has an article for Arithmetic billiards.

This morning I tried to write a draft concerning a proof of infinitude of primes: the idea is translate to the language of arithmetic billiards the idea of the proof of infinitude of primes by Filip Saidak ([1], I don't know the article really I know [2] in Spanish that refers the article); if I'm right my idea was evoke an infinitude of arithmetic billiards embedded/nested in each other (you can think that the first of them has dimensions $\text{base}\times\text{height}$ equal to $2\times 3$ and the following billiard $6\times 7$… following Saidak's recipe to get the proof), and combine this construction with Bézout's identity and the definition of greatest common divisor and an inductive/recursive argument.

Question. Provide some novel example, or a novel proof of a well-known theorem (theorems at graduate level) or a novel result (at research level) in mathematics arising from an arithmetic billiard of your invention.

From [3] I know more applications of arithmetic billiards to mathematics. I wondered if we can to get more applications, I mean novel results, of arithmetic billiards in mathematic (new proofs of known theorems or novelty and curious results at research level, as soon there are some contributions I should accept an answer).

As you see, I evoked an infinitude of arithmetic billiards with the goal to prove Euclid's theorem of the infinitude of primes (I don't know if one can prove that there are infinitely many primes with a more elegant way using arithmetic billiards). You can use thus an arbitrary number of arithmetic billiards, you can use also if you want/need it a non-Euclidean geometry, you can use a different shape for your billiard(s) in the dimensions that you need for your construction and argument. In your arithmetic billiards you can use balls or rays of light (and you can evoke the reflection or refraction laws),… or other suitable requirements that you need to evoke in your construction.

The references [2] and [3] written by professor Bartolo Luque are in Spanish from the journal Investigación y Ciencia that is the Spanish edition of Scientific American.

I add that this week I've edited a post in Mathematics Stack Exchange, with identifier 4497455 and title Particular values for the sum of divisors function from billiards (thus I add the link here as companion of my post in MathOverflow, feel free to explore this kind of question if you're interested and you know how to compute particular values of multiplicative functions divisors functions, or as the Dedekind psi function see Wikipedia, Euler's totient function or in general arithmetic functions related to analytic number theory.

References:

[1] Filip Saidak, A New Proof of Euclid's Theorem, The American Mathematical Monthly 113(10), pages 937-938 (2006).

[2] Bartolo Luque, La hipótesis de Riemann (I), Investigación y Ciencia, Diciembre 2020 Nº 531, pages 83-87.

[3] Bartolo Luque, El billar como computador analógico, Investigación y Ciencia, Mayo 2021 Nº 536, pages 86-90.

Answer 1

This answer IS A DRAFT to add some ideas about a prime constellations, the near square primes (I didn't exploit the theory of Euler's totient function in the context of visible points, I think that this arithmetic function is related to this subject; I didn't exploit other arithmetic functions as the Dedekind function in the context of lattices). I don't add the explanation of the figures because my English is bad, but I add a conjecture and notation.

Wikipedia has a section dedicated to near-square primes from the article Landau's problems.

I add the file https://i.sstatic.net/lO3Km.jpg that shows (I notice that the drawings are poor) the figure F1 and the figure F2. I believe that the following conjecture holds, where $\varphi(n)$ denotes the Euler's totient function.

Conjecture. Denote $p=\square+1$ a near-square prime. The integer $\square+1$ is a near-square prime if and only if $$\varphi(\square+1)=\frac{1}{\sqrt{2}}\operatorname{length}(\gamma)\times\#\{\text{distinct singular clocks}\}\tag{1}$$ holds.

The notation (isn't motivated) is $G$ denoting the gnomon for the small square unit below the square $p-1=\square$ that's represented in figure F1, $\gamma$ is the trajectory of these singular arithmetic billiards (are singulars in some way because the starting point of the path in red colour from the gnomon is through a corner. The asterisk denotes that the singular clock (this specific singular arithmetic billiard) is repeated in our inventary of singular arithmetic billliards, and two asterisks denote that the last billiard is an arithmetic billiard that isn't singular, really it is an ordinary or regular arithmetic billiard and we remove this billiard from our inventary.

Notice that $48=\varphi(64)\neq 64$, thus the integer $\square+1=8^2+1$ isn't a counterexample for our Conjecture.

The figure F2 is just to add a failed attempt to characterize again near-square primes with arithmetic billiards. Denote the side $l=6$ our near-square prime is $p=\square+1=l^2+1=37$, then $2l-1=11$, and with the figure I tried to combine the well-known formula $(2l-1+2l-3)+\ldots+(7+5)+(3+1)=l^2$, billiards and particular values of Euler's totient function (I emphasize that I don't know how to get a characterization of near-square primes from figure F2). In a figure likes than F2 ($2\cdot 6-1$ rows and $6$ columns) I have drawn the fibers in colour green and red denoting these with a capital letter F, these fibers have lengths $(2k-1)\sqrt{2}$ for certain integers $k\geq 1$ (see the mentioned formula in this paragraph). I tried to think about quotients from the nodes that I have drawn in figure F2, but I believe that it doesn't makes sense to get a characterization of near-square primes using these fibers and billiards/particular values of Euler's totient function or Dedekind psi function.