A Derivation of the Infinitude of Primes

AbstractThe well-known analogy between polynomials and integers breaks down when it comes to considering the polynomial derivative. This is rather unfortunate since derivatives are a powerful tool for doing arithmetic with polynomials. Nevertheless, there are some proposals in the literature for arithmetic analogues of derivatives. In this article we use one of these arithmetic derivatives to give a proof of the infinitude of primes which is analogous to an argument that will be presented for polynomials using polynomial derivatives. We hope that this “differential” proof of the infinitude of primes will help to motivate the reader to look for good notions of arithmetic derivatives.MSC: 11A4111C08 ACKNOWLEDGMENTThis article was possible thanks to the support of ANID Fondecyt Regular Grants 1190442 and 1230507 from Chile. I thank Ricardo Menares and Natalia Garcia-Fritz for valuable feedback on a first version of this article. I also thank the referees and editors for numerous suggestions.Additional informationNotes on contributorsHector PastenHECTOR PASTEN is a Chilean mathematician. He studied at Universidad de Concepción under the supervision of Xavier Vidaux (Chile, 2010) and at Queen’s University under the supervision of Ram Murty (Canada, 2014). Then he was a Benjamin Peirce Fellow at Harvard University (2014-2018). During this period he was also a visiting scholar at the Institute for Advanced Study at Princeton (2015-2016). In 2018 he returned to Chile where he is now an Associate Professor at the Mathematics Department of Pontificia Universidad Católica de Chile. He is interested in number theory and its connections with mathematical logic and analysis.