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 have dimensions $\text{basis}\times\text{ height}$ 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 novelty example, or a novelty proof of a well-known theorem (theorems at graduate level) or a novelty result (at research level) in mathematics arising from an arithmetic billiard of your invention. Many thanks.

From [3] I know more applications of arithmetic billiards to mathematics. I wondered if we can to get more applications, I mean novelty 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 to 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 to prove that there are infinitely many primes with a more elegant way using arithmetic billiards). You can to use thus an arbitrary number of arithmetic billiards, you can to use also if you want/need it a non-Euclidean geometry, you can to 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 to use balls or rays of light (and you can to 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 identificator 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 questions 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.

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.

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 have dimensions $\text{basis}\times\text{ height}$ 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 novelty example, or a novelty proof of a well-known theorem (theorems at graduate level) or a novelty result (at research level) in mathematics arising from an arithmetic billiard of your invention. Many thanks.

From [3] I know more applications of arithmetic billiards to mathematics. I wondered if we can to get more applications, I mean novelty 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 to 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 to prove that there are infinitely many primes with a more elegant way using arithmetic billiards). You can to use thus an arbitrary number of arithmetic billiards, you can to use also if you want/need it a non-Euclidean geometry, you can to 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 to use balls or rays of light (and you can to 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.

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.

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 have dimensions $\text{basis}\times\text{ height}$ 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 novelty example, or a novelty proof of a well-known theorem (theorems at graduate level) or a novelty result (at research level) in mathematics arising from an arithmetic billiard of your invention. Many thanks.

From [3] I know more applications of arithmetic billiards to mathematics. I wondered if we can to get more applications, I mean novelty 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 to 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 to prove that there are infinitely many primes with a more elegant way using arithmetic billiards). You can to use thus an arbitrary number of arithmetic billiards, you can to use also if you want/need it a non-Euclidean geometry, you can to 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 to use balls or rays of light (and you can to 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 identificator 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 questions 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.