A Derivation of the Infinitude of Primes
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Date
2024
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Abstract
The 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.
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11A41, 11C08