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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.

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  • $\begingroup$ If you billiard uses refraction laws you can to define the different mediums of refraction just providing a sequence for the refraction indexes (that correspond to different material of your billiard). $\endgroup$
    – user142929
    Jun 16, 2022 at 12:42
  • $\begingroup$ I add also that I'm trying to refresh subjects of non-Euclidean geometry. for example I'm interested in pages 46-47 from Introductory Non-Euclidean Geometry, by Henry Parker Manning, Dover Publications Inc (2005). These seem to me interesting in the way to try to compute in different way particular values of arithmetic functions (I hope that my ideas to build/define alternative billiards aren't charlatanism); I know also the article Dennis P. Walsh, A curious Way to Test for Primes, Mathematics Magazine, Vol. 80, No. 4 (Oct., 2007), pp. 302-303. $\endgroup$
    – user142929
    Jul 23, 2022 at 14:14
  • $\begingroup$ All users: I will try to edit in the next few days on Mathematics Stack Exchange a question about billiards for primes represented as a L (and the shape [ representing two odd prime numbers and the shape $\sqcup$ for the prime L and its specular reflection, ...). These billiards are placed inside or outside a box, in front of walls or mirrors,... I provide certain representations (the previous that I've evoked in this comment) invoking Goldbach conjectures, and simple formulas involving the Euler's totient function. (At my home I call these gadgets prisms and pendulums.) $\endgroup$
    – user142929
    Aug 9, 2022 at 15:39
  • $\begingroup$ Since this is a post tagged as (recreational-mathematics) tomorrow if I can to digitalize the images, I add the identifier of a file from Imgur with recreational designs for sand clocks (three examples), clepsydra/water clocks (two examples of such arithmetic billiards) and my attempt from four billiards to get an ammonia clock (while this last arithmetic billiard it's an unfinished attempt, thess examples of artihemtic billiards seem to me very nice). All these example harmonize with the examples of billiards and prime numbers of my edited posts. Many thanks for your patience and help. $\endgroup$
    – user142929
    Dec 9, 2022 at 17:25
  • $\begingroup$ The sand clocks are the images in figure F1 from the Imgur file with identifier s050cTz where S in the figure denotes the starting point; the water clocks are in the file from Imgur with identifier lUnqWmD and finally my attempt to define an ammonia clock evoking "tunneling quantum" is in the file from Imgur with identifier bZzZ9fO , again S denotes the starting point of trajectory, in case of the molecule NH$_3$ at right of figure F3 the length of the trajectory is greater than the lengths of trajectories of the molecules at left. $\endgroup$
    – user142929
    Dec 10, 2022 at 13:01

1 Answer 1

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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.stack.imgur.com/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.

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  • $\begingroup$ I've added in capital letters and bold that this answer is a draft, since I didn't explain the figure F1, if you're able to explain in English the figure F1 and get a proof of Conjecture please edit this post-Community Wiki, many thanks. $\endgroup$
    – user142929
    Sep 8, 2022 at 11:04
  • $\begingroup$ If the post is well-received I will try to add in comments my thoughts about other singular billiards (and gnomon) for other constellations of primes, and sure that I'm going to add as companion of this "answer" a comment about formulas involving near-suare primes, but I can not motivate these with billiards. Many thanks for your patience and feedback. $\endgroup$
    – user142929
    Sep 8, 2022 at 11:06
  • $\begingroup$ (1/2) Claim. Let $A=B^2+1$ be a prime number, then the following equations hold with $\sigma(n),\varphi(n)$, $\psi(n)$ and $\operatorname{rad}(n)$ denoting, respectively, the sum of divisors function, the Euler's totient function, the Dedekind psi function and the radical of an integer: $\sigma(A)=B^2\cdot\frac{\psi(A)}{\varphi(A)}=\frac{\varphi(A)\psi(A)}{B^2}$; $\varphi(\varphi(A))=\sqrt{A-1}\cdot\varphi(B)=\frac{1}{\sqrt{-1+A/B^2}}\cdot\varphi(B)$; $\endgroup$
    – user142929
    Sep 8, 2022 at 12:36
  • $\begingroup$ (2/2) $\varphi(B)\cdot\operatorname{rad}(\varphi(A))=B\varphi(\operatorname{rad}(B))$; $\varphi(B)\psi(\varphi(A))=\psi(B)\varphi(\varphi(A))$; $B\psi(A-1)=(A-1)\psi(B)$; $B\varphi(A-1)=(A-1)\varphi(B)$ and $\sqrt{A}\varphi(B)=\sqrt{\varphi(B)^2+\varphi(\varphi(A))^2}$ that also can be written as $\sqrt{A\varphi(B)^2-\varphi(\varphi(A))^2}=\varphi(B)$. (I hope that there aren't typos, if you know some of these formulas from the literature or an attempt to get a characterization/interpretation by using billiards of near-squares primes from these formulas please add a comment.) $\endgroup$
    – user142929
    Sep 8, 2022 at 12:37
  • $\begingroup$ In previous Claim the integer $B$ is greater than $1$, that's $B>1$. $\endgroup$
    – user142929
    Sep 8, 2022 at 12:46

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