Irracionalidade recíproca
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916296037236736 |
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| author | Isneri, Renan Jackson Soares Vieira, Vandenberg Lopes da Silva, Maxwell Aires |
| author_facet | Isneri, Renan Jackson Soares Vieira, Vandenberg Lopes da Silva, Maxwell Aires |
| contents | Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to guarantee the storage of data and the sending of messages in a secure way. And this is evident in e-commerce when personal data must be kept confidential. The proof that $\sqrt{p}$ is an irrational number, for every positive prime $p$, is known, if not by everyone, at least by the majority of Mathematics students, and such a proof is, in general, given by means of a basic property of numbers primes: if $p$ divides the product of two integers, then it divides at least one of them. This result forms the basis of other equally important results, such as, for example, what is given by the Fundamental Theorem of Arithmetic, which is the basic result of the Theory of Numbers. In this article, we present a proof of the irrationality of $\sqrt[2n]{p}$ using results from Quadratic Residue Theory, especially, by Gauss's Law of Quadratic Reciprocity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10338 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Irracionalidade recíproca Isneri, Renan Jackson Soares Vieira, Vandenberg Lopes da Silva, Maxwell Aires History and Overview Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to guarantee the storage of data and the sending of messages in a secure way. And this is evident in e-commerce when personal data must be kept confidential. The proof that $\sqrt{p}$ is an irrational number, for every positive prime $p$, is known, if not by everyone, at least by the majority of Mathematics students, and such a proof is, in general, given by means of a basic property of numbers primes: if $p$ divides the product of two integers, then it divides at least one of them. This result forms the basis of other equally important results, such as, for example, what is given by the Fundamental Theorem of Arithmetic, which is the basic result of the Theory of Numbers. In this article, we present a proof of the irrationality of $\sqrt[2n]{p}$ using results from Quadratic Residue Theory, especially, by Gauss's Law of Quadratic Reciprocity. |
| title | Irracionalidade recíproca |
| topic | History and Overview |
| url | https://arxiv.org/abs/2307.10338 |