Irracionalidade rec\'iproca

TL;DR

This paper presents a novel proof of the irrationality of the 2n-th root of a prime number using quadratic residue theory and Gauss's Law of Quadratic Reciprocity, expanding the mathematical understanding of prime-related irrational numbers.

Contribution

The paper introduces a new proof of the irrationality of 9nth roots of primes based on quadratic residue theory and Gauss's Law, differing from traditional proofs.

Findings

Proof of irrationality of 9nth roots of primes.

Application of quadratic residue theory and Gauss's Law.

Enhanced understanding of prime-related irrational numbers.

Abstract

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.